A Quantum Computer Foundation for the Standard Model and Superstring Theories*

A Quantum Computer Foundation for the Standard Model and SuperString Theories*

Stephen Blaha**

* Excerpted from the book Cosmos and Consciousness by Stephen Blaha (1stBooks Library, Bloomington, IN, 2000) available from amazon.com and bn.com. ** Associate of the Harvard University Physics Department. ABSTRACT

1. SuperString Theory can naturally be based on a Quantum Computer foundation. This provides a totally new view of SuperString Theory.

2. The Standard Model of elementary particles can be viewed as defining a Quantum Computer Grammar and language.

3. A Quantum Computer can be represented in part as a second-quantized Fermi field.

4. A Quantum Computer in a certain limit naturally forms a Superspace upon which Supersymmetry rotations can be defined – a Continuum Quantum Computer.

5. A representation of Quantum Computers exists that is similar to Turing Machines - a Quantum Turing Machine. As part of this development we define various types of Quantum Grammars.

6. High level Quantum Computer languages are described for the first time.

New linguistic views of the most fundamental theories of Physics, the Standard Model and SuperString Theory are described. In these new linguistic representations particles become literally symbols or letters, and particle interactions become grammar rules. This view is NOT the same as the often-expressed view that Mathematics is the language of Physics. The linguistic representation is a specific new mathematical construct.

We show how to create a SuperString Quantum Computer that naturally provides a framework for SuperStrings in general and heterotic SuperStrings in particular.

There are also a number of new developments relating to Quantum Computers and Quantum Turing Machines that are of interest to Computer Science.

The elementary particle language defines a new form of reality in the sense that the elementary particles become states within a Quantum Computer. The workings and contents of the universe are reduced to computer computation. Speculations about the purposes and goals of the elementary particle language are presented based on the goals and features of languages like C++ and Java.

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iii justified. It is known to be incomplete because it does not include gravitation.

The Standard Model started with the unification of the theory of Quantum Electrodynamics (electromagnetism) with the theory of the Weak interactions in a theory that is called the Electroweak Theory. 11 This theory was developed by Steven Weinberg, Shelley Glashow, Abdus Salam and others. Afterwards the Electroweak theory was combined with the theory of the Strong interactions to produce the The Standard Standard Model. The Standard Up to this point we have been sketchy in our description of the fundamental particles of Nature because there is a widely accepted Model of belief amongst physicists that today’s “fundamental particles” may not Model of be fundamental in a deeper theory of matter. Elementary The Families of Matter Elementary Today the fundamental particles of nature, as described by the Standard Model, are grouped into three families: fermions, gauge field Particles particles and Higgs field particles. Fermions are particles of half- Particles integer spin that constitute what we ordinarily call matter. The fermions include electrons, quarks, muons, neutrinos and so on. Our understanding of the fundamental nature of matter today is in one way much more developed than that of 1900. In another way we still Gauge field particles are particles that “carry” the interactions or are at a stage where we think of particles of matter as fuzzy little balls. forces between particles. You could call them “force field” particles The balls are different – quarks, leptons, and so on in year 2000 because they embody the forces between particles. The photon is a instead of atoms of chemical elements in year 1900. And we know gauge field. There are other gauge fields for the weak interaction and more about their properties. But they are still fuzzy little balls. for the strong interaction. The accepted theory of the day is called the Standard Model1. It is an Higgs field particles are heavy spin zero particles that are needed in amalgam of earlier theories of electromagnetism, the weak interaction the current formulation of the theory. They have not as yet been and the strong interaction. The Standard Model is consistent with all observed. The reason is thought to be their high mass. Particle known experimental results. This combination of existing theories accelerators are only now becoming capable of producing them. There works but it has a number of arbitrary features that cannot be logically are numerous speculations that Higgs particles are not fundamental but may be some form of composite particle. Since they have not been 1 There are many books on the Standard Model. See for example Kerson Huang’s book, Quarks, Leptons & Gauge Fields (World Scientific Publishing, River Edge, observed experimentally at the time of this writing there is no way of NJ, 1992). knowing. 1 The fermion family has two subfamilies: the quarks and the leptons. The sets of color triplets are distinguished from each other by a One distinguishing feature of quarks vs. leptons is that quarks can quantum number called the flavor quantum number. Each triplet of experience the strong force but leptons cannot. The lepton family quarks, and a corresponding lepton, has the same flavor quantum includes electrons, muons () and neutrinos (). The members of the number. fermion family are listed in the following table: The classification of quarks and leptons in the preceding fermion table The Fermion Family is the result of over fifty years of experimental and theoretical analysis. Generation Flavor Quarks Leptons It is like the periodic table of the elements that is so important for I 1 up u1 u2 u3 e electron neutrino chemistry. 2 down d1 d2 d3 eelectron

II 3 charmed c1 c2 c3 muon neutrino 4 strange s1 s2 s3 muon The Standard Model incorporates all of our current experimental III 5 top t1 t2 t3 $ tau neutrino knowledge of particles and interactions (except gravity) in a single 6 bottom b1 b2 b3 $ tau theory. The theory has been extremely successful in accounting for many experimental results although many interesting physical Notice also the fermions appear in three generations or sets labeled I, calculations in this theory cannot be done with current computational II, and III. There is no obvious reason for this repetition of techniques (such as the binding of light quarks to produce protons, generations. We do not know at present why Nature has three neutrons, and so on). In fact, many physicists feel the Standard Model generations rather than one generation. It is like having three sets of agrees too well with the available experimental data. china when one set would do. Each generation contains 6 quarks and two leptons. The lack of discrepancies and disagreements with experiment is a negative in a sense. Disagreements between experiment and theory are Notice that each quark comes in three colors. The colors are labeled usually the source of advances in our understanding of nature. with the subscripts 1, 2 and 3. For example the “down” quark actually Grappling with discrepancies leads to new theoretical ideas. Like the comes in a triplet – three varieties – that we have denoted d , d , and d 1 2 3 ideal marriage with no arguments the success of the Standard Model in the above table. These triplets are called color triplets. The variety without experimental discrepancies makes life less interesting for the of quark colors is tied up with the fact that they experience the Strong physicist. force. Current Problems of the Standard Model Originally the three varieties of each quark type were labeled with While the Standard Model is not in clear disagreement with any colors such as red, white and blue. (A “red” quark was not actually current experimental data there are a number of problems that the red. The name only served to distinguish between the quark varieties.) Standard Model does not resolve as far as we know. Currently these Now we use simple numeric subscripts. In fact these labels reflect problems are: internal quantum numbers that are called color quantum numbers by physicists. Consequently, the theory of the Strong Interactions of the 1. Explaining the “dark matter” found in galaxies quarks is often called Quantum Chromodynamics since it involves the Astronomers have found that the observed galaxies in the universe contain color quantum numbers. normal matter and an unidentified form of matter that has been given the name “dark matter”. Dark matter is undetectable except through its gravitational effects. The amount of dark matter in the universe appears to

2 be about the same as the amount of normal matter. Without dark matter, galaxies would not have enough matter to hold them together and would fly 2. The “weird” pattern of symmetries in the Standard Model apart because the force of gravity due to visible normal matter is The Standard Model contains a set of symmetries SU(3) for Quantum insufficient. Dark matter has the interesting feature that it experiences the Chromodynamics, and SU(2) and U(1) for the Electroweak part that are not gravitational force but it does not experience any of the other forces as far easy to justify. These symmetries U(1), SU(2), and SU(3) are symmetry as we know. Light and cosmic rays (elementary particles) appear to pass groups chosen from a vast (infinite) number of possibilities appearing in a through dark matter as if it was empty space. Dark matter has some of the branch of Mathematics called Lie Groups after their originator Sophus Lie. interesting properties of the ether of a hundred years ago. A number of Why Nature chose these groups is completely unclear. Ideally the “correct” schemes have been proposed to account for dark matter. The absence of theory would contain one symmetry group that we could somehow justify experimentally verifiable predictions makes it hard to judge whether the on physical grounds. This “correct” theory would then, through some correct answer has been found. mechanism, split into these separate symmetries in the approximate theory called the Standard Model. SuperString Theory is one attempt to create such 2. Incorporating gravity into the Standard Model a theory. The Standard Model is a Quantum Field Theory. Adding the Theory of Gravity as developed by Einstein and others to the Standard Model results Despite these problems the Standard Model is an extremely successful in a theory that generates infinitely large numbers (and an infinite number of them at that) when calculations are performed. The normal quantization theory and a triumph for the physicists – experimental and theoretical of the Theory of Gravity as a Quantum Field Theory produces a theory that – whose labors over the last seventy years led to its creation. cannot make finite predictions for perfectly ordinary quantum phenomena and instead produces meaningless infinities. The inability of the Standard In subsequent sections of this chapter we will look at some of the Model to incorporate the gravitational interaction without creating interactions between particles in the Standard Model. The interactions difficulties may be a sign of the need for a new theoretical approach. involve the creation and absorption of particles. We will also look at these interactions from a linguistic point of view in keeping with the 3. The appearance of “too many” arbitrary constants in the Standard general theme of this book that Theoretical Physics is in reality a Model theory The Standard Model contains at least 19 arbitrary constants. These numbers search for a language with which to describe Nature. The approach have values that we can determine experimentally but their appearance in followed here will tie in with the discussions of quantum computers in the theory is unsatisfying. Physicists hope that there is some deep reason succeeding chapters. that these numbers have the values found experimentally. This hope leads to the idea that the Standard Model is a precursor to a deeper theory of reality. Interactions of the Standard Model The Standard Model consists of a complex mathematical expression 1. The lack of “elegance” in the Standard Model called a lagrangian together with a procedure for quantization, and a The Standard Model has been described as three theories lumped together: Quantum Electrodynamics, the Theory of the Weak Interaction, and the procedure for the calculation of physically interesting quantities. The Theory of the Strong Interaction (called Quantum Chromodynamics). quantization and calculation procedures are specified by Quantum Although the Electroweak (Quantum Electrodynamics combined with the Field Theory. The details of these procedures are important but they Weak Interactions) part of the Standard Model has some intertwined are not relevant to the theme of this book. structure, Quantum Chromodynamics is simply “tossed into the pot” to create the combined theory. An elegant theory would be a closely interconnected theory with all the parts intertwined in a significant way. The lagrangian for the Standard Model can be divided into two parts. The close interconnections are a missing ingredient that fills physicists with One part describes the behavior of non-interacting particles. These misgivings about the Standard Model and leads them to view the Standard particles are called free particles. After some mathematical Model as an interim theory. 3 development we can think of free particles as moving through space in details that are not central to the theme of this book.) This term and interacting with other free particles through the exchange of gauge represents a number of different interaction situations. The situations particles. Gauge particles are the particles that are the carriers of the can be depicted graphically as: electromagnetic, weak, and strong forces. The photon is the gauge particle for Quantum Electrodynamics (electromagnetism). Three W bosons are the gauge particles for the Weak Interaction. (The photon and the W particles are intertwined in the Electroweak theory.) Eight gluons are the gauge particles for Quantum Chromodynamics (the Strong Interaction).

The second part of the lagrangian describes the interactions of Electron Emits Photon Electron Absorbs Photon particles. These interactions typically involve the exchange of gauge particles between the interacting particles. The interactions on the quantum scale become the forces we see in everyday life when vast numbers of quantum interactions take place.

It is an interesting point of history to note that hundreds of years ago scientists thought that light was made of particles and that seeing involved the reception of particles. Forces were also viewed as the result of myriads of particles acting on a body. These ideas were discarded in the nineteenth Photon Annihilates into Electron and Positron century only to be revived in the latter part of the twentieth century in a Electron and Positron Annihilate into photon much deeper and more detailed quantitative form. Figure. Simplest Electromagnetic Interactions of an electron. Solid The interaction part of the Standard Model specifies many interactions lines are electrons and positrons (anti-electrons). Dashed lines are between particles. We will look at a representative sample of these photons. interactions to obtain an understanding of their basic idea and then use these interactions to illustrate the linguistic representation of particle The interactions depicted in the figure correspond to an amazing interactions. The linguistic approach will be explored in more detail in feature of matter: particle creation and annihilation with the succeeding chapters. conversion of energy into matter and vice versa. This energy- matter conversion requires both the quantum theory and the theory of Special The first interaction term in the lagrangian that we will explore can be Relativity to work. Special Relativity is required because of the written in the form: conversion between matter and energy. The creation and annihilation process is an inherently quantum process requiring Quantum Field eAe theory.

where e represents an electron (the bar over the left e signifies an The interaction processes in the preceding figure are the simplest “outgoing” electron in our simplified view) and A represents a photon. electromagnetic interactions of an electron. These simple interactions (Many significant features are omitted to avoid getting bogged down can be combined to make an infinity of more complex composite interactions. For example two electrons can scatter off each other by 4 exchanging a photon. The photon exchange takes place by having one calculations are normally visualized using Feynman diagrams. For electron emit a photon and the other electron absorb it. This composite example the collision of two electrons to produce two electrons with process can be pictured with a (Feynman) diagram such as: different energy and momenta can be visualized as a sum of terms corresponding to the various ways the electrons can interact. This sum of terms is an approximate perturbation theory calculation. It is approximate because the exact answer would require calculating an infinite number of terms – and nobody knows how to perform such a calculation. The simplest and dominant terms in electron-electron scattering are:

= + + …

Figure. Feynman diagram for two electrons interacting by exchanging a photon (dotted line). Figure. The first few terms in the approximate perturbation theory The preceding discussion shows how the simple interactions in the calculation of the scattering of two electrons. The dotted lines Standard Model lagrangian can produce a complex variety of represent photons that carry the electromagnetic force between the phenomena. The Feynman diagrams that we have seen are used electrons. extensively in particle physics. They are named after their originator Richard Feynman. These diagrams correspond to very complex rules The individual terms in the perturbation theory calculation can be for performing calculations in quantum field theory. The calculations viewed as words. The words are part of a language with an alphabet produce probability numbers giving the probability that a given and grammar. process will take place. The 36+ particles of the Standard Model constitute the set of symbols Perturbation Theory and a Language for the Standard or the alphabet of the language. (It is an interesting coincidence that Model most alphabet-based human languages have 20 to 40 letters in their This brief view of some of the features of the Standard Model is a alphabets. English has 26 and so on.) As we shall see, the grammar is preliminary to a new view or representation of the physics of the quantum extension of a type of computer language (developed by Standard Model. In this view the interaction terms in the lagrangian Chomsky and others) that uses production rules. The production rules specify the grammar of a computer language. for the grammar of the Standard Model are easily derived from the interaction terms of the Standard Model. Physicists must use an approximation procedure called perturbation theory to perform computations in the Standard Model. Perturbation The concept of the language of the Standard Model is very simple. The theory uses the interaction terms of the lagrangian to perform an technical details of the language description require a discussion of approximate calculation of particle interactions. Perturbation theory computer languages and Quantum Turing Machines. The next few 5 chapters describe the basic idea of the linguistic representation of the Standard Model. They show how particle interactions can be viewed as transformations (processes) within a Quantum Computer that accepts the computer language generated by the Standard Model lagrangian interaction terms.

6 particles to make an input string. This input string undergoes transformations specified by grammar rules to produce an output string of letters corresponding to the outgoing particles after the collision.

In describing this new representation of the Standard Model we will focus on the essentials of the processes of creation, transmutation and 22 annihilation of matter ignoring (for the moment) particle spin, momentum, angular momentum and other details that are important in the complete theory of the Standard Model. (These details will be considered when we consider the SuperString Quantum Computer. TThhee LLiinngguuiissttiicc Incorporating them into a language is not difficult.) The idea of associating physics with computers is not as unconventional as it might appear at first. Feynman2 viewed computers RReepprreesseennttaattiioonn ooff as relevant for Physics: “If we suppose we know all the physical laws perfectly, of course we don’t have to pay any attention to computers. It’s interesting anyway to entertain oneself with the idea that we’ve got tthhee SSttaannddaarrdd MMooddeell something to learn about physical laws; and if I take a relaxed view here … I’ll admit that we don’t understand everything.” Feynman wanted to simulate physics computations on a quantum computer in In the previous chapter we explored the features of the Standard the hope that it would be faster than a conventional computer. We will Model. We saw that it was consistent with all the known properties of show the Standard Model itself actually defines a specific (theoretical) elementary particles except for gravitation, and phenomena such as quantum computer – a far more exciting possibility – because it gives dark matter. This chapter introduces a new view or representation of a new view of Reality. Nature itself is a form of computer. We will the Standard model that focuses on its interactions and shows the also show that SuperString theory can be formulated within a Quantum Standard Model defines a language similar to a computer language. Computer framework.

The alphabet of the language is the set of elementary particles of the A computer language representation of particle physics is of great Standard Model. The words of the language are quantum states interest in itself. It may generate new insights into the process of consisting of elementary particles. A bound state of several particles matter creation and transformation. It may lead to a new understanding such as a proton (three quarks bound together) is a word. A quantum of the fundamental nature of the universe. And it appears to suggest a state consisting of several free particles – particles that are not bound rationale for approaches such as the currently popular SuperString together and that are some distance from each other also constitute a theories of elementary particles. word. In fact the entire universe constitutes one mighty word.

We shall see that the collision or scattering of particles can be viewed as beginning with a combination of letters corresponding to the 2 R. P. Feynman, International Journal of Theoretical Physics, 21, 467 (1982). 7 Linguistic View of an Interaction They contain quantum numbers and other properties (mass, spin, and We will begin by looking at the simple interaction term we explored in so on) that certainly are data. the previous chapter: Computer Grammars eAe The Standard Model lagrangian in our view specifies a grammar in the sense of Naom Chomsky. Chomsky’s concept of a language, and of a From a computer language perspective this lagrangian interaction term grammar, has important applications in the theory of computation and can be viewed as specifying a set of grammar rules called production computers. rules. There are four basic types of languages in the Chomsky approach: type In fact each interaction term in the Standard Model lagrangian can be 0, type 1, type 2 and type 3. They differ in the forms of the grammar viewed as specifying a set of grammar rules. The combined set of rules (also called production rules) that they allow. We will be grammar rules for all Standard Model interaction terms constitutes a interested in type 0 languages. A type 0 language (also called an grammar equivalent to the Standard Model lagrangian interaction unrestricted rewriting system) is the most general type of language. It terms with particles becoming the alphabet (letters or symbols) of the allows any grammar production rule of the form grammar. x y To appreciate the mapping or analogy of particles and alphabetic letters, and of interaction terms and computer grammar, we have to where x and y are strings of characters. understand the process of data characters (or letters) flowing through a computer. It is an interesting and little noted fact (because it is viewed Production rules specify how one string of characters transforms into as trivial) that a computer can generate (or absorb) data as part of the another string of characters. Calculations in computers using computer computation process. For example we might write a computer program languages are reducible to sets of grammar rules for string that takes a set of letters input into a computer and outputs each input manipulation that are similar to the one shown above. letter twice: Each term in the interaction part of the Standard Model lagrangian is equivalent to one or more production rules where the characters are abc computer aabbcc particles. The Standard Model can be viewed as generating a type 0 language. This language goes beyond current types of grammars because it is inherently quantum probabilistic in nature. Quantum In a sense the computer creates data characters just like interactions aspects of these rules will be described later. can create particles. Computers can also absorb (or annihilate) data (usually to our dismay). So we can see an analogy between the Before looking at the production rules generated by an interaction term transformations of data characters in a computer, and particle in a lagrangian we will look at a formal grammar. A grammar is a annihilation and creation. Nothing else in Nature is so directly quadruple of items that is symbolized by the expression analogous to particle creation and annihilation. We take the view that particles are bundles of data that we can denote with letters (symbols).

8 where N is a set of variables called nonterminal symbols, T is a set of terminal symbols, S is a special nonterminal symbol called the head or Application start symbol, and P is a finite set of production rules. The angle braces of String of String of < and > are merely a mathematician’s way of saying these items are terminals production terminals grouped together to constitute (or make) a grammar. rules

The terminal symbols are the set of characters that are allowed in input Figure. Generating an output string of terminal characters from an strings or output strings. The nonterminal symbols are the set of input string of terminal characters using the production rules of a characters that appear in intermediate steps that lead from the input grammar. Inside the black box transformations of the input string take string to the output string. They are like internal variables or symbols. place and non-terminal symbols may appear and disappear. Non- The combined set of terminal and nonterminal symbols make up the terminal symbols, by definition, cannot appear in the input or output vocabulary (alphabet) of a language. strings of characters.

Chomsky’s definition of a language is the set of all strings of terminal In order to make these grammar concepts more concrete we will look symbols that can be generated by applying the production rules to the at a simple artificial grammar before looking at the grammar generated head symbol (or start symbol) S. The head symbol is the symbol that by interaction terms in the Standard Model. The nonterminal symbols begins all strings of symbols that can be generated in the language. will be the letters S (the head symbol), A and B. The terminal symbols will be the letters x and y. The production rules will be A simple example of a language in this approach is a vocabulary or alphabet consisting of the ABC’s with words created from these letters S → AB Rule I according to some set of production rules. A → y Rule II A → Ay Rule III We will generalize Chomsky’s idea of language to be the set of all B → x Rule IV strings that can be generated from all input strings of terminal symbols B → Bx Rule V as well as the head symbol. We can also view all particles as generated directly or indirectly at the beginning on the universe. The “Big Bang” The computer language that this grammar generates consists of all (the beginning of the universe) then becomes the primeval head strings containing any number of y’s followed by any number of x’s symbol. since any of these strings can be generated from the head symbol S using the production rules. Because of rule I y’s are placed to the left A visualization of the application of production rules to transform an and x’s are placed to the right. The order of the symbols matters just as input string of terminal characters into an output string of terminal it does in human language – consider the words ides and dies which characters is: differ only in the order of i and d!

An example of generating a string yyxxx from the head symbol is:

9 S → AB Rule I AB → AyB Rule III AyB → yyB Rule II yyB → yyBx Rule V yyBx → yyBxx Rule V yyBxx → yyxxx Rule IV

The production rule used to make each transition is listed on the right.

Our generalization of the Chomsky definition of language would allow any string to be the starting point – not just the head symbol S. Using the sample grammar described on the previous page the generalized language becomes any string of x’s and y’s. To make it more interesting we can add two new rules:

y → A Rule VI x → B Rule VII

The language – the set of strings of terminal symbols – remains the same despite the addition of these new grammar rules. However the number and variety of transitions becomes much larger. For example the following chain of transitions is allowed,

yx → AB → AyBx → AyyBx→ AyyyBxx → yyyyxx

10 The head symbol is the letter S. The terminal symbols are the letters x and y. The production rules are:

S → xy Rule I x → xx Rule II Relative Probability = .75 x → xy Rule III Relative Probability = .25 33 y → yy Rule IV The probability of generating the string xxy vs. the probability of Probabilistic generating the string xyy from the string xy is Probabilistic xy → xxy relative probability = .75 CCoommppuutteerr vs. GGrraammmmaarrss xy → xyy relative probability = .25 The string xxy is three times more likely to be produced than the string xyy. Probabilistic Computer Grammars For each starting string one can obtain the relative probabilities that The preceding chapter described the production rules for a various possible output strings will be produced. deterministic grammar. The left side of each production rule has one, and only one, possible transition. A more practical example of a Probabilistic Grammar can be abstracted from flipping coins – heads or tails occur with equal Non-deterministic grammars allow two or more grammar rules to have probability – 50-50. From this observation we can create a little the same left side and different right sides. For example, Probabilistic Grammar for the case of flipping two coins. Let us let h represent heads and t represent tails. Then let us choose the grammar: A → y A → x S → hh → could both appear in a non-deterministic grammar. S tt S → ht → Non-deterministic grammars are naturally associated with S th probabilities. The probabilities can be classical probabilities or h → t probability = .5 (50%) quantum probabilities. An example of a simple non-deterministic h → h probability = .5 (50%) grammar is: t → h probability = .5 (50%) 11 t → t probability = .5 (50%)

The last four rules above embody the statement that flipping a coin yields heads or tails with equal probability (50% or .5). Detectors 1 Now let us consider starting with two heads hh. The possible outcomes Source and their probabilities are: 2 hh → hh probability = .5 * .5 = .25 hh → th probability = .5 * .5 = .25 hh → ht probability = .5 * .5 = .25 hh → tt probability = .5 * .5 = .25

If we don’t care about the order of the output heads and tails then the Figure. Two slit photon experimental setup. probability of two heads hh → ht or th is .25 + .25 = .5. A simple Quantum Probabilistic Grammar can be constructed This simple example shows the basic thought process of a non- corresponding to this experimental setup: deterministic grammar with associated probabilities. S → 1 probability amplitude = 1/√2 → √ The combination of a non-deterministic grammar and an associated set S 2 probability amplitude = 1/ 2 of probabilities for transitions can be called a Probabilistic Grammar. We will see that the grammar production rules for the Standard Model The head symbol S represents the source. The digit 1 represents a must be viewed as constituting a Probabilistic Grammar™ with one photon going through slit 1. The digit 2 represents a photon going difference. The “square roots” of probabilities – probability amplitudes through slit 2. are specified for the transitions in the grammar. Probability amplitudes are required by the Standard Model because it is a quantum theory. The values of the probability amplitudes 1/√2 can be calculated using Therefore we will call the grammar of the Standard Model a Quantum Quantum Mechanics. The probability for a photon to go through slit 1 Probabilistic Grammar™. is the absolute value squared of the probability amplitude:

2 Quantum Probabilistic Grammar Probability to go through slit 1 = (1/√2) = .5 An example of a Quantum Probabilistic Grammar™ can be constructed in analogy with a two slit photon experiment. Imagine a and the probability for a photon to go through slit 2 is wall with two slits. A source of photons shoots photons at the wall. A photon can go though either slit with equal quantum probability. An Probability to go through slit 2 = (1/√2)2 = .5 illustration of this experimental arrangement is: This simple example illustrates the basics of a Quantum Probabilistic Grammar™.

12 Before applying these concepts to the Standard Model we will look at The simplest string transition from a two & “input” state to a one & a simpler Quantum Field Theory called a &3 (“phi cubed”) theory (& is “output” state is: the Greek letter phi). This theory describes a self-interacting spin 0 particle with no internal symmetries. This artificial theory is a stepping && → & stone to the far more complex Standard Model Quantum Field Theory. using Rule II. The probability amplitude of this transition is g by The &3 theory is so named because it has a lagrangian interaction term assumption. with that form. The grammar rules for the &3 theory are: The transitions between strings can be visualized with diagrams that & → && Rule I are like the Feynman diagrams that used in Quantum Field Theory && → & Rule II perturbation theory calculations. These diagrams are not the same as Feynman diagrams because they embody time orderings of emissions Rule I corresponds to the emission of a & particle and rule II and absorptions of & particles. corresponds to the absorption of a & particle. We do not introduce a start symbol. Instead we will consider the transitions from an input The time order of emission and absorption of & particles can be state of a number of & particles to an output state of possibly a symbolized using parentheses. For example, different number of & particles. We will ignore the momenta of the particles. (This assumption is equivalent to infinitely massive (&)& → (&&)& = &(&&) → &(&) = && Diagram A particles.) We will assume either transition above takes place with a and “relative probability amplitude” g. We will call this simplified theory the modified &3 theory. We will view g as a measure of the probability &(&) → &(&&) = (&&)& → (&)& = && Diagram B amplitude for an absorption or emission of a &. (g is similar to a coupling constant in Quantum Field Theory.) The actual summed These string transitions correspond to different time-ordered Feynman- probability has to be normalized or rescaled so that the sum of like diagrams: probabilities equals one. A B To get a feel for the Quantum Probabilistic Grammar™ approach we & && & & & will look at the case of an input state of two & particles. The output = + states can have one &, two &’s, three &’s, and so on. Each possible & && & & & output state has a certain probability of occurring. The sum of the Feynman diagram Feynman-like diagrams with different time orderings probabilities of producing all output states must equal one. (Remember that the sum of all possible outcomes of flipping a coin is one. Having Figure. The true Feynman diagram is the “sum” of the two time- it come up heads has probability ½ and having it come up as a tails has ordered Feynman-like diagrams. probability ½ also. And ½ plus ½ equals one.) The correspondence between the Feynman-like diagrams and the transitions between strings based on the Quantum Probabilistic 13 Grammar™ can be seen by taking vertical slices on diagrams A or B above after each emission or absorption. For example, We will now look at the transition of two & particles to two & particles: && → &&. There are an infinite number of Feynman-like diagrams for A B this transition.

= + Now in evaluating diagrams to calculate the probabilities we must remember that we are ignoring space-time aspects such as particle propagators. So the calculation of the probability amplitude for this 1 2 3 1 2 3 Feynman diagram Feynman-like diagrams with different time orderings process becomes a counting of the number of diagrams that exist for each power of g2. The probability amplitude for each diagram is a Figure. Slicing Feynman-like diagrams. A slice is made after each power of g. emission or absorption. As you read down a slice the particles are listed in the same order as the corresponding string. Each slice is Counting diagrams is a combinatorial mathematics problem that we numbered. will not explore in detail because it is peripheral to our interests. Consequently we will simply write the probability amplitude as: The string corresponding to each slice in the above figure is numbered ∞ in the transitions: 2n A2(g) = an g n=1 1 2 3 (&)& → (&&)& = &(&&) → &(&) = && A where the mathematical expression on the right represents a sum from n = 1 to infinity and where the numbers an are integer numbers equal to 2n and the number of different diagrams having a power of g . Each intersection of lines (called a vertex) contributes a factor of g to the 1 2 3 amplitude for that diagram. The powers of g for the simpler diagrams &(&) → &(&&) = (&&)& → (&)& = && B are: g2 g4 g4 The parentheses on the left side of the arrow indicate the particle that emits the new particle appearing within the parentheses on the right = + + side of the arrow.

4 4 A transition from an input state containing & particles to an output g g state containing & particles always has an infinite number of ways of taking place and thus an infinite number of Feynman-like diagrams. + + + … Readers familiar with perturbation theory in Quantum Field Theory can see that these diagrams are the same as the Feynman diagrams generated by perturbation theory with the additional feature of having The value of the first constant a1 is 1 since there is only one Feynman 2 time orderings. diagram with value g for this process – the first diagram to the right of 14 the = sign above. We will treat all time-ordered variations of a where the numbers bn count the number of distinct diagrams with the 2n+1 Feynman diagram as contributing one to the value of an. The value of power g and an grows rapidly as n increases. For large n the value of an is of the order of (n!)2. Consequently the sum is an asymptotic power series. 2 P3 = |A3(g)| Here again the details are not important. We are not looking for a numerical result. The absolute (normalized) probability to produce an n & particle output state is The (unnormalized) relative probability for the transition && → && is: Q = P /P 2 n n P2 = |A2(g)| The sum of all possible output state probabilities equals one: In quantum theories the probability is always the square of the absolute value of a probability amplitude. The probability P2 is a relative probability that must be normalized – multiplied by a factor ∞ that makes the sum of the probabilities of all possible outcome states equal to one. To calculate this probability we must calculate the sum 1 = Qn n=1 of all the relative probabilities Pn to produce any number of & particles from a two & input state. The modified &3 Quantum Field Theory provides a simple example of && → & … a Quantum Probabilistic Grammar™. We will now turn to the Standard Model and examine its Quantum Probabilistic Grammar™. The total of the relative probabilities is: Because it encompasses a much larger number of different particles (letters) and interactions (grammar rules) it will be significantly more ∞ complicated. P = Pn n=1 where Pn is the relative probability to produce an output state with n & particles.

The calculation of the relative probabilities Pn for n & particles output states is similar to the calculation P2. For example,

∞ 2n+1 A3(g) = bn g n=1

15 pe → A A → ep A → pe

where e represents an electron, p represents a positron, and A represents a photon. The production rules describe the emission and 44 absorption of photons by electrons and positrons as well as the annihilation of an electron and positron to produce a photon, and the decay of a photon into an electron-positron pair.

An example of an interaction between two electrons in the linguistic SSttaannddaarrdd MMooddeell approach is:

1 2 3 QQuuaannttuumm ee → eAe → ee

where the electrons interact by exchanging one photon. One Feynman- GGrraammmmaarr™™ like diagram for these transitions is:

Grammar Production Rules of Electromagnetism We will start by considering electromagnetic interactions in the Standard Model as grammar production rules. The production rules for the electromagnetic interaction term for electrons and positrons in the Standard Model eAe are: e → eA e → Ae eA → e Ae → e p → pA p → Ap Ap → p pA → p ep → A 16 1 2 3 e e 1 2 3 e e

A A ee ee

Figure. A diagram showing how two electrons interact by exchanging a photon. As time increases the electrons move from left to right. The Figure. Another diagram with two electrons interacting by exchanging upper electron emits the photon. This corresponds to the left e a photon. In this case the lower electron emits the photon. This transitioning to eA using the grammar rule e → eA. corresponds to the right e in the initial ee string transitioning to Ae using the grammar rule e → Ae. The vertical slices in the Feynman diagram which are numbered 1, 2 and 3 correspond to the three numbered strings in the transitions The “normal” Feynman diagram for this process represents the sum of generated from the production rules. Each string has an ordering that both of the previous diagrams: corresponds to the order of particles as you descend a slice. For example slice 2 has an electron, photon, and another electron in that = + order as you descend corresponding to string 2 above.

Another Feynman-like diagram that is important and contributes to Feynman diagram Feynman-like diagrams with different time orderings this process has the lower electron emitting a photon that is then absorbed by the upper electron. Figure. The normal Feynman diagram represents several of our Feynman-like diagrams with different time orderings of particle emission and absorption.

A more complex example of a Feynman-like diagram is:

17 1 2 3 4 5 6 absorbed the photon.) This ambiguity reflects the fact that there are several possible time orderings. The preceding diagram actually corresponds to eA(Ae) → eA(e). The right electron absorbs the photon.

To calculate the probability of an actual physical transition taking place such as ee → eepe we must take account of all diagrams with all possible time orderings for the specified input and output states.

The above examples show how the electromagnetic interaction part of the Standard Model lagrangian can be viewed as defining a grammar. The grammar has corresponding Feynman-like diagrams. These types of diagrams are not without precedent in Physics. They have time orderings that are similar to the time orderings that appeared in Figure. A diagram for the collision of two electrons that produce a new perturbation theory calculations around 1950. electron-positron pair: ee → eepe. The Weak Interaction and Strong Interaction parts of the Standard Six slices appear corresponding to various intermediate states in this Models also define grammars. Consequently we can view the complete complex electron-electron interaction. The production rules can be Standard Model lagrangian as defining a grammar where the “letters” used to generate a sequence of strings that correspond to the slices: (alphabet or vocabulary) are the elementary particles of the model and the Feynman-like diagrams corresponding to the Standard Model can 1 2 3 4 5 6 be viewed as a sequence of strings generated by applying the ee → eAAe → eAe → eAAe → eepAe → eepe production rules specified by the Standard Model lagrangian.

As you descend each slice the particles are ordered in the same way as Production Rules for the Weak and Strong Interactions the corresponding string. For example, as you descend slice 5 the order The Weak and Strong interaction terms in the Standard Model of the particles is electron, electron, positron, photon and electron. lagrangian are also easily translated into grammar production rules. We will illustrate these cases using the Weak interaction terms: The transitions of character strings have an ambiguity. For example - the above transition νeW e and 2 3 eAAe → eAe - νµW µ → could have taken place through (eA)Ae (e)Ae with the left e ν ν → where e represents an electron neutrino, µ represents an muon absorbing an A or through eA(Ae) eA(e) with the right e absorbing µ - an A. (Parentheses are used for grouping to show which electron neutrino, represents a muon, W is a gauge field of the Weak interaction and e is an electron; and the Strong interaction term 18 - and so on where e is an electron, p is a positron, W is a negative W uGu gauge boson, ν is a neutrino, G is a Strong interaction gauge boson and u and u’ are u quarks which may have different color quantum where u is a u quark and G represents gauge fields of the Strong numbers. interaction. These production rules generate Weak interaction transitions such as Notice that neutrinos come in different varieties: electron neutrinos, muon decay: muon neutrinos and tau neutrinos. The three kinds of neutrinos have different internal quantum numbers that distinguish them. Neutrinos 1 2 3 do not have electromagnetic charge. They are neutral as their name µ → - ν → ν ν suggests. Each kind of neutrino has a corresponding charged partner. W µ e e µ We are familiar with the electron. The other charged partners are the muon and tau particle. These particles are like heavy electrons for the with the corresponding Feynman-like diagram: most part. The three charged leptons have different internal quantum numbers. 1 2 3 µ νµ The above interaction terms imply production rules such as:

→ - ν e W e - W - e → νe W e - W → e νe νe - W → νe e - They also generate Strong interactions between quarks with color µ → νµ W gauge fields:

- µ → W νµ 1 2 3 uu → uGu → u’u’ u → Gu’

’ with the corresponding Feynman-like diagram: u → u G

19 The character string transition approach based on production rules is 1 2 3 equivalent to the Feynman diagram approach. It does however provide u u’ a different, and simpler, view. Interestingly the Feynman diagram approach to calculations is very difficult and tedious – but it often leads to a simple result due to the massive cancellation of many G complex terms with each other. (An attempt to simplify perturbation theory diagram calculations was made by Wu and Cheng in the early uu’ 1970’s (and by this author privately). The simple linguistic approach might be a hint of a more efficient way of calculating in field theories like the Standard Model where the complications are absent from the very beginning.

Whether or not the linguistic approach leads to a less complicated The preceding examples are the simplest cases of the infinite variety of theory we have now reached a rather amazing result. After 2500 years Feynman-like diagrams that can be generated from the Standard of speculation on the nature of matter we have developed a Model production rules. surprisingly simple theory (for everything except gravitation) called the Standard Model that can be viewed as a type 0 computer language. The Standard Model Quantum Grammar It has an alphabet (vocabulary) of roughly 36 or so particles, and a set At this point we have created a view of the Standard Model of of production rules specified by the interaction part of the Standard elementary particles in which the particles are an alphabet of at least Model lagrangian. thirty-six letters (eighteen quarks, six leptons, twelve gauge fields, plus possibly other particles such as Higgs bosons). This situation represents something of a miracle. There is no reason that Nature should have so few particles that interact with each other The particle alphabet can be combined into strings that represent input through a simple set of rules. A computer language theorist would call or output states of scattering particles. Transitions between strings take the language of the Standard Model a language with a finite place through the grammar production rules and correspond to time- representation. Simply put, this means the words of the language can ordered Feynman-like diagrams. be generated from a finite vocabulary (alphabet or set of particles) and a finite set of production rules. So we now have a language, the Standard Model, with letters, words and a grammar, and an interpretation of the language in terms of rules The Standard Model Language is Surprisingly Simple of calculation and experimental setups. Many physicists feel that there are too many elementary particles in the Standard Model. From the point of view of a language theorist the The linguistic representation of the Standard Model that we have finite language representation of the Standard Model is a very special developed omits many important calculation details as well as situation. As Hopcroft and Ullman3 point out, “there are many more important properties of particles such as spin and particle momenta. These features could be added in a direct way. The focus of our investigation is on the essentials of the act of creation of particles in 3 J. E. Hopcroft and J. D. Ullman, Formal Languages and Their Relation to particle interactions. Automata, (Addison-Wesley, Reading, MA, 1975) page 2. 20 languages than finite representations.” Languages can have infinite alphabets or infinite sets of production rules or other complications.

The physical equivalent of an infinite alphabet would be a universe with an infinite number of different types of matter. Every particle of matter could have a different mass and differ in other properties. From this point of view the Standard Model is truly a marvel of simplicity.

The Standard Model, in fact, is a very compact, finite description of most of the known features of our universe. The linguistic view of the Standard Model suggests we should view elementary particles as symbols or clumps of data – a vocabulary. The interactions of the elementary particles involve the creation or annihilation of particles – creation in the deepest form seen by man.

Remarkably, the data flowing through a computer can be viewed as being transformed from one form to another and being output to different destinations. Data flowing through a computer can be divided into streams that can be sent to different output channels such as a printer or the computer screen. For example, the character, a, can be a data item in a stream of data that is sent to both the printer and the screen: printer a computer screen

The streaming of data characters to different output channels is very analogous to the output of particles from particle scattering as we have seen.

The simple production rules that describe particle scattering in the Standard Model suggest a fundamental simplicity at the core of Reality that goes far beyond the speculations of philosophers and scientists of earlier ages.

21 Turing machine can recognize and analyze so that it can process the data to produce output.

The language of the Standard Model is a type 0 computer language. A type 0 language requires a Turing machine to handle its productions. Because particle transitions are quantum and because the left side of a 55 production rule can have several possible right sides (For example, a photon can transition to an electron-positron pair, a quark-antiquark pair and so on.) the Turing machine for the Standard Model language must be a non-deterministic Quantum Turing Machine.

QQuuaannttuumm TTuurriinngg Features of Normal Turing Machines Before examining a Quantum Turing Machine for the Standard Model we will look at the features of “normal” Turing machines. A normal MMaacchhiinneess Turing machine consists of a finitely describable black box (its features are describable in a finite number of statements) and an infinite tape. The tape plays the role of computer memory. The tape is What are Turing Machines? divided into squares. Each square contains a symbol or character. The The linguistic view of the Standard Model leads to a number of character can be the “blank” character or a symbol. A tape contains questions. One important question is the nature of the Turing machine blank characters followed by a finite string of input symbols followed that accepts this language. A Turing machine is a generalized by blank characters. theoretical computer that is often used to analyze computational questions in computer science. The black box consists of a control part and a tape head. The control part has a finite set of rules built into it (the “program”) and a finite A personal computer can be viewed as a special purpose Turing memory that it uses as a scratch pad normally. The tape head can read machine. A personal computer has memory in the form of RAM and symbols from the tape one at a time and can move the tape to the left, hard disks. A personal computer has built-in programs that tell it what right, or not move it, based on instructions from the control part. to do when data is input into the computer. Similarly a Turing machine also has memory and instructions within it telling it how to handle The tape head tells the control the symbols it is scanning from the tape input and how to produce output from a given input. and the control decides what action to have the tape perform based on When we type input on a computer keyboard or have input come from the scanned symbols and information (the program and data) stored in another source such as the Internet or a data file, the input has to be in the control’s memory. a form that the computer can handle. Similarly, the input for a Turing machine must have a specific form for the Turing machine to accept it, process it and then produce output. In the case of a Turing machine we say the input must be presented in a language that the Turing machine “accepts”. In this context the word “accepts” means a format that the

22 calculable in the Standard Model. We need a Quantum Turing Black Box Machine to handle this more complex situation. Control Part Quantum (and non-quantum) Turing Machines can be pictured in a Finite set of Rules Finite Memory convenient way by viewing the control part as containing a tape on which the rules are inscribed, the current state of the Turing machine is Tape Head specified, and the current symbol being scanned by the tape head is stored.

The grammar rules of a Quantum Turing Machine are quantum probabilistic. In the simplest case each grammar rule has an associated number that we will call its relative probability amplitude. We will call this type of Quantum Probabilistic Grammar™ a factorable Quantum … b i n p u t b … Probabilistic Grammar. The calculation of the probability for a transition from a specified input string to a specified output string is based on the following rules: Tape 1. The relative probability amplitude for an input string to be Figure. Schematic diagram of a Turing machine. transformed to a specified output string is the sum of relative probability amplitudes for each possible sequence of transitions A set of input symbols is placed on the tape and the rules (program) in that leads from the input string to the output string. the control part are applied to produce an output set of symbols. 2. The relative probability amplitude for a sequence of This process is analogous to elementary particle processes: an input set grammar rule transitions is the product of the relative of particles interacts through various forces of nature and produces an probability amplitudes of each transition. output set of particles. The difference is that elementary particle processes are quantum probabilistic in nature. The laws of Physics 3. The relative probability for an input string to be transformed (which appear to be finitely describable since they can be specified by to an output string is the absolute value squared of the relative the Standard Model lagrangian except for gravity) play the role of the probability amplitude for the input string to be transformed to finite set of rules. the output string.

Quantum Probabilistic Grammars 4. The absolute probability for an input string to be The major difference between Turing machine outputs and the outputs transformed to a specified output string is the relative in particle physics are that output states in particle physics are probability for the input string to be transformed to the quantum probabilistic. A given set of input symbols (particles) can specified output string divided by the sum of the relative produce a variety of output states with different probabilities probabilities for the input string to be transformed to all

23 possible output strings using the grammar rules. This rule A general Quantum Probabilistic Grammar is an entangled Quantum guarantees the sum of the probabilities sums to one. Probabilistic Grammar in which we cannot assign a probability amplitude to individual grammar rule transitions. The relative The &3 theory example considered previously is an example of a probability amplitude for a sequence of grammar rule transitions is not factorable Quantum Probabilistic Grammar™. The grammar rules for the product of the relative probability amplitudes of each transition. the &3 theory example were: Instead in an entangled Quantum Probabilistic Grammar the relative Transition Relative Probability Amplitude probability amplitude for a sequence of grammar rule transitions is a & → && g function of the combined set of grammar rule transitions. && → & g The Standard Model grammar is in fact an entangled Quantum The relative probability amplitude for Probabilistic Grammar if we take account of the momenta and spins of the input and output particles. An illustration of the entanglement in input state: && → output state: && the Standard Model is electron scattering through the exchange of two photons. The Feynman diagram for this process with particle momenta was shown to be displayed is: ∞ p + k 2n A2(g) = an g p p’ n=1

in the preceding chapter. The relative probability for this process is: k p – p’ +k 2 P2 = |A2(g)| q q’ and the absolute probability for this process was shown in the q - k preceding chapter to be An electron with momentum p collides with an electron with Q2 =P2/P momentum q and exchanges two photons (dotted lines). The momentum of each electron and photon is displayed. The outgoing where electrons have the momenta p’ and q’. The probability amplitude for ∞ this diagram has a factor of e4 (a power of e for each of the four P = Pn transitions or vertices that generate or absorb a photon). This factor is n=1 similar to the probability associated with a transition in factorable with Pn being the relative probability for a two & particle input state to Quantum Grammars™. In addition there is a factor associated with the become an n & particle output state. overall pattern of transitions. The resulting probability amplitude is proportional to

24 e4¶ d4k N(k) k2[(p+k)2 – m2] [(q-k)2 – m2] [(p–p’+k)2 – m2] where N(k) is a numerator that depends on the electron momenta, spins, and the “loop” momentum k that is being integrated over. The integral depends on the details of the arrangement of the transitions and thus provides the entanglement. The calculation of probability amplitudes in Quantum Field theories is described in many books on the subject.

We conclude that the Quantum Probabilistic Grammar™ of the Standard Model is an entangled Quantum Probabilistic Grammar.

25 Tape I contains the specification of the grammar rules expressed as a finite string of symbols, the current state of the Quantum Turing Machine, and other data. Tape II contains the input string. After applying the grammar rules in a quantum probabilistic way an output state is generated. The output state is placed on Tape II in the simplest 66 Quantum Turing Machine implementation.

TThhee SSttaannddaarrdd Control MMooddeell QQuuaannttuumm Tape Head

CCoommppuutteerr Tape I

current … aaaaa Finite String of Rules Current State symbol The Standard Model Quantum Turing Machine scanned The Quantum Turing Machine that corresponds to the Standard Model has a number of exciting features that distinguish it from conventional Turing Machines. Tape II … bbbbbb[input state]bbbbb … First it accepts a language that has a finitely describable entangled Quantum Grammar™. Although the Standard Model has an entangled Figure. Quantum Turing Machine. Tape I plays the role of computer Quantum Grammar the grammar rules are finitely describable. Finitely memory. Tape II is memory for input and output. describable means that the rules can be specified by a finite set of symbols. The rules generated from the interactions of the Standard The behavior of a Quantum Turing Machine can be viewed as: Model are finite in number and each rule consists of a finite number of symbols. Thus the rules generated from the Standard Model are 1. The Turing machine begins in the input state specified on tape II. finitely describable. An input string is placed on tape II. The other memory locations on tape II are filled with blank characters. In our case this string is a A Quantum Turing Machine can be visualized as consisting of a list of symbols for an input set of elementary particles that are control element and two tapes that play the role of computer memory. about to interact. The Turing machine we are considering accepts The control has a tape head that reads and writes symbols to the two any state consisting of a finite number of elementary particles. The tapes. 26 connection between Turing machines and computer languages is probability of being produced that can be calculated using the brought out at this stage. A machine “accepts” a language if it can Standard Model. take any sentence (set of particles in our case) of the language, and perform a computation producing output (a set of output particles 3. The Quantum Grammar™ rules of the Standard Model Quantum in our case). A Turing machine that accepts a language is an Turing Machine have internal symmetries that result in symmetries embodiment of the grammar of the language. in the input and output states.

2. The Quantum Turing Machine applies the grammar rules to the 4. Since the momenta and spins of the input and output particles are input set of states in all possible ways to produce an output state physically very important the Standard Model Quantum Turing that is a quantum superposition of states. Each possible output state Machine must take account of these properties in the input and has a certain probability of being produced. output states as well as internally when calculating transition probabilities. 3. The probability for producing a specified output state from a specified input state can be calculated as we illustrated in a simple So we must picture the input particle state on tape II as containing not example earlier using the relative probabilities associated with the only the particle symbol but also momenta and spin data. Quantum Grammar™ rules of the modified &3 theory. To get an idea of how a Quantum Turing Machine would take an input 4. The set of possible states of a Quantum Turing Machine4 is infinite set of particles and produce a set of output particles we will consider unlike non-Quantum Turing Machines that only have a finite the case of two electrons colliding with such energy that an electron- number of possible states. positron pair is created:

The Standard Model Quantum Turing Machine has some distinctive ee → eepe features: where e represents an electron and p represents a positron (the 1. Since the order of the particles in the input state string is not electron’s antiparticle). One of the corresponding Feynman-like physically important we will consider the input string to be diagrams is: actually all permutations of the order of the particles in the input.

2. Since the Turing machine is quantum the rules are probabilistic in nature: a given set of input particles will in general produce many possible output particle states. Each output state will have a certain

4 D. Deutsch, Proceedings of the Royal Society of London, A 400 97 (1985) describes (universal) Quantum Computers and points out they can simulate continuous physical systems because they have a continuum (infinite number) of possible states. As page 107 points out “a quantum computer has an infinite- dimensional state space”. Quantum Computers are equivalent to Quantum Turing Machines as we will see. 27 produced is a sum of the contributions of all the possible alternate ways of reaching that output state. 1 2 3 4 5 e e In addition, electron-electron scattering can produce many other output states depending on the initial energy of the electrons. Each output state has its own probability of occurring. Some examples are:

γ ee → eqqe e ee → eepepe

ee → eµµe

p ee → eµµepe γ where q represents a quark, µ represents the muon antiparticle and q represents an antiquark. e e The Quantum Turing Machine representation does raise several interesting prospects for the theory of elementary particles embodied where γ represents a photon. The input string can change according to in the Standard Model. First, the Quantum Turing Machine the grammar rules in the following way: representation raises the possibility that some of the powerful techniques and general results of the theory of computation can be ee → eγe → eγγe → eepγe → eepe brought over to physics and perhaps provide guidance on the next stage after the Standard Model. Since the Quantum Turing Machine is probabilistic there are many – in fact an infinite number – of ways in which the transition Secondly, and perhaps more importantly, the separation of the input and output states (they are on tape II) from the intermediate ee → eepe calculational states of the Turing machine (that are on tape I) is suggestive of a somewhat different approach to the fundamentals of can take place – each with its own probability of happening. The particle interactions: The space-time of the incoming and outgoing sample sequence of transitions shown above is only one of these particles may be different from the “space-time” describing the possible ways. The total probability of this transition is the square of interactions and internal structure of the interacting particles. This the sum of the probability amplitudes for all possible ways according view is based on thinking of tapes I and II as representing separate to quantum mechanics. Nature requires us to take account of all space-times. possible ways of transitioning from the input state of particles to the output state of particles. The total probability of the output state being

28 A precursor of this point of view appears in Quantum Field Theory. In Quantum Field Theory the interaction of particles is viewed as consisting of three phases: an initial state where the particles are widely separated and distinct, an interaction region where the particles “collide” and interact perhaps creating new particles, and a final state where the outgoing particles are widely separated.

Figure. Two particles collide and generate a three particle outgoing state in Quantum Field Theory.

In conventional Quantum Field Theory the space-time in the interacting region is conventionally assumed to be the same as the space-time of the incoming and outgoing states. Nevertheless Quantum Field Theory distinguishes the interaction region from the region of the incoming and outgoing particles.

The SuperString approach to the theory of elementary particles introduces a separate space-time to describe the elementary particles. Elementary particles are viewed as strings vibrating in this space-time. Can one view the SuperString space-time as tape I and the external behavior of the elementary particles taking place in normal space-time as tape II? Perhaps the Quantum Turing Machine representation of the Standard Model is the key to the next level of our understanding of elementary particles and Nature.

29 Feynman’s Quantum Computer was particularly adapted to simulating physical quantum systems. He gave a concrete picture of it as a space- time lattice that has two possible states at each point on the space-time 77 lattice.

QQuuaannttuumm x CCoommppuutteerrss aanndd

FFoocckk SSppaaccee t

Figure. A Quantum Computer viewed as a space-time lattice of points. Up to this point we have been looking at Quantum Turing Machines. At each intersection point there are two possible states: occupied or Recently much excitement has been generated by Quantum unoccupied. Alternately we can use a spin ½ description and say that Computers. A Quantum Computer is an alternate and earlier we have spin “up” or spin “down” at each space-time point (indicated formulation of the Quantum Turing Machine concept. In 1982 Richard with up arrows or down arrows). Or we can say that a computer bit Feynman5 described a new theoretical concept called a Quantum exists at each space-time point that can have the value 1 or 0. Computer. Quantum Computer concepts had been developed by Benioff6, Deutsch and others in preceding years. Feynman then hypothesized that every finite quantum mechanical system could be exactly described (exactly simulated) with such a A Quantum Computer can be viewed as similar to a “normal” system if the system were treated as a finite lattice of interacting spins computer in many ways. However it is not deterministic. It transitions and a suitable interaction was “chosen” between the spins at the from one state to another in a quantum probabilistic way. adjacent lattice points. (We are interested in infinite lattices.)

The Quantum Computer would start with the spins on the lattice set up to correspond to an initial quantum state of the quantum system being simulated. Then the interaction between the spins would cause 5 R. P. Feynman, International Journal of Theoretical Physics, 21, 467 (1982). transitions in the spins mirroring the evolution of the quantum system 6 Paul Benioff, Jour. Stat. Phys. Vol 22, p.563 (1980); Jour Stat. Phys. Vol 29, p 515 (1982); Phys. Rev. Letters Vol 48, p 1581 (1982); Int. Jour Theoret. Phys. Vol 21, being modeled. The final state of the Quantum Computer would 177 (1982). correspond to the final state of the quantum system being simulated. In 30 particular the probability of a transition to a specified output state of The mapping or correspondence between the Quantum Turing the Quantum Computer would be identical to the probability of a Machine and a Quantum Computer of the type of Feynman and transition of the quantum system being modeled to its corresponding Deutsch is based on the following correspondences: output state. Symbol → number → bit pattern → lattice spins Feynman’s concept of a Quantum Computer was concrete and physical – you could view it as a lattice of spins – an idea that Each symbol in the vocabulary of a Quantum Turing Machine can be physicists were comfortable with. Since then, numerous individuals given a numerical value just as alphabetic characters in a real have explored theoretical aspects of quantum computers including D. computer are given numerical values internally. The numeric value of Deutsch.7 Deutsch provided a more general formulation of Quantum each symbol can be written in base 2 as a binary string of zeros and Computers based on quantum bits which he called a universal quantum ones just as modern computers do. The string of zeros and ones computer. His universal quantum computer is capable of “perfectly specifies bits in a Quantum Computer. The initial quantum state of a simulating every finite, realizable physical system.” Recently, Quantum Computer is the combined set of bits corresponding to the quantum computers have been shown to offer significant advantages symbols in the input string of symbols of the corresponding Quantum over normal computers in some types of calculations8. Turing Machine.

The concept of the Quantum Computer as developed by Feynman and The Quantum Grammar of the Quantum Turing Machine maps to an others is an alternate representation of the Quantum Turing Machine interaction between the spins on the lattice of the Quantum Computer. concept. The Quantum Turing Machine concept is based on a The interaction between the spins must correspond to the Quantum linguistic approach with a Quantum Grammar, and a vocabulary of Grammar™ in the sense that, for any given input state, the output state symbols. We chose to develop this representation of Quantum of the Quantum Computer must correspond to the equivalent output Computers because it was the natural approach for the Standard state of the Quantum Turing Machine. Model. The Standard Model interaction lagrangian specified the Quantum Grammar with a vocabulary consisting of the fundamental Another way of stating this correspondence between the Quantum particles of the Standard Model. Grammar™ and the lattice spin interaction is that the probability of producing a specified output state from a specified input state must be Can we map our Quantum Turing Machines to Quantum Computers the same for a Quantum Turing Machine and its equivalent Quantum and show they are equivalent concepts? Actually this mapping is quite Computer. The Quantum Grammar must correspond to an equivalent easy to do. interaction between the spins on the Quantum Computer lattice.

Quantum Grammar ↔ Interaction between spins of lattice

Thus we have established a mapping between Quantum Turing 7 D. Deutsch, Proceedings of the Royal Society of London, A 400 97 (1985) and A Machines and Quantum Computers. 425 73 (1989). P. Benioff explored state descriptions even earlier. See footnote 6. 8 P. W. Shor, Algorithms for Quantum Computation: Discrete log and Factoring, Bell Laboratories paper, 1998. Many other excellent papers on the advantages of Quantum Computers have appeared since Shor’s seminal paper. 31 The Continuum Limit of a Quantum Computer – A The spin “dimension” parameter of the Continuum Quantum Superspace Computer can be viewed as a Grassmann variable parameter. To A Quantum Computer has a lattice of points. At each point there is a illustrate this correspondence let us imagine that we construct a bit that can be in an “up” or a “down” state. Let us imagine quantum wave function )(x) that specifies the probability amplitude to “shrinking” the lattice so the space between lattice points becomes find “spin up” vs. “spin down” at each space-time point. Let smaller and smaller. Eventually the lattice approaches what could be called the continuum limit where the points form a continuous space. At each point of this space there is a bit that can be either “up” or )1(x) “down” and, as a result, we can view this space as a superspace )(x) = similar to that encountered in Supersymmetry theories. We will call )2(x) this limiting case of the Quantum Computer a Continuum Quantum Computer™. be the (Weyl) spinor wave function for the quantum computer in the continuum limit. The component )1(x) is the probability amplitude At each lattice point the spin is either up or down. As the lattice that the spin is “up” at the point x, and the component )2(x) is the spacing shrinks we encounter discontinuities in the spin. The probability amplitude that the spin is “down” at the point x. We discontinuities form a countable set of measure zero, i.e. they can be assume )(x) is continuous except possibly for a set of measure 0. counted using the set of integers. We smooth over these discontinuities to have smoothly varying spins in the continuum limit of the lattice. )(x) can be rewritten in terms of spinors as: The information contained in the smoothly varying spins is the

evolving data in the Quantum Computer. )(x) = )1(x)s1 + )2(x)s2 where In Supersymmetry theories9 there is a parameter space consisting of space-time coordinates (parameters) which we will denote as x , and 1 Grassmann variable parameters denoted and . Grassmann s1 = parameters are not normal variables because they satisfy anti- 0 commutation rules. and 0 The Continuum Quantum Computer also has an extended space-time. s2 = It has a space-time parameter x together with a spin at every space- 1 time point. The index is a number specifying coordinate. For 0 ordinary space-time ranges from 0 to 3. The coordinate x is the time are spinors. Now the Grassmann variable has a Weyl spinor index 1 2 3 coordinate. The coordinates x , x and x are the space coordinates. as well. So we can represent our wave function as

)(x) = )1(x)1 + )2(x)2

9 We will follow the approach of D. Bailin and A. Love, Supersymmetric Gauge Thus our Continuum Quantum Computer maps directly to a space of Field Theory and String Theory (Institute of Physics Publishing, Philadelphia, PA, the type required for Supersymmetry. 1994) pages 13, 23, and 34. 32 original Continuum Quantum Computer. (We assume that edge effects Now we take the Supersymmetric transformations and reinterpret them – effects of the edge of the Continuum Quantum Computer’s space are in terms of the physics of the Continuum Quantum Computer. In a negligible.) Supersymmetric space the form of a Supersymmetric “rotation” is: SuperSymmetric Continuum Quantum Computer x t x + a - i" + i" If we now imagine creating a Continuum Quantum Computer that is equivalent to a Quantum Turing Machine for an enhanced theory of t + elementary particles that includes SuperSymmetry, then its space will be of infinite extent. We will call this computer the SuperSymmetric t + Continuum Quantum Computer. (We can also create a Continuum Quantum Computer for SuperString theory as we show in a later where a specifies a shift (translation) in space-time coordinates, and chapter.) and are constant anti-commuting Grassmann parameters. A Supersymmetric “rotation” can shift the space-time coordinates and The entire universe can be viewed as contained in the SuperSymmetric inter-mix them with the Grassmann parameters. The -i" + i" Continuum Quantum Computer™. The lagrangian for that theory can expression is an ordinary numeric expression – not an anti-commuting be added to the “edge” of the universe as the rules (or program) part of Grassmann value. The Supersymmetric “rotation” is reminiscent of the the quantum computer forming an augmented universe. rotation between space and time that we see in the Theory of Special Relativity. Only here, in Supersymmetric space, it rotates between The lagrangian for the universe that we add to the “real” universe can ordinary space-time coordinates and Grassmann coordinates. be thought of as occupying a universe of its own. Spaces or universes of lagrangians have been discussed by Kenneth Wilson and others. An The interpretation of the Supersymmetric “rotation” from the interesting question raised by the existence of a space of lagrangians is viewpoint of the Continuum Quantum Computer™ is quite interesting. whether a principle exists in the space of lagrangians that can be used To understand it we must first realize that the information in the to deduce the lagrangian that Nature seems to implement just like there memory of a Quantum Computer is not only the bits in the memory is a way to deduce laws of Physics from the Standard Model lattice. The distribution of the bits – how they change as one goes lagrangian. One possibility suggested by the computer framework of through memory is also part of the information contained in the the discussion is a lagrangian that is generated through a self- Quantum Computer’s memory. With that in mind we can view the organizing process of the sort envisioned in Artificial Intelligence Supersymmetric transformation that intermixes space-time and the studies. This possibility remains to be explored. Grassmann spinor parameters as a rotation between the data bits in the Continuum Quantum Computer and the space-time of the Continuum This representation of the enhanced SuperSymmetric Continuum Quantum Computer. Quantum Computer effectively reduces Nature to Language. Everything in the universe including the laws of Physics becomes part If the spin interactions of the Continuum Quantum Computer™ are of a quantum computer. The space of the quantum computer is filled invariant (unchanged) under a Supersymmetric transformation then the with bit patterns describing the data and “program” or rules of the transformed Continuum Quantum Computer is equivalent to the computer.

33 Group Symmetries of Quantum Computers Only 6 bits are actually required for the tightest, most minimal It is interesting to note that the type of symmetries of Quantum representation of approximately 36+ particles. The actual number of Computers that we have been discussing, and their implications for bits and the role of each bit could be chosen to provide the most computation, has not been addressed in the literature on Quantum elegant representation of a particle’s internal quantum numbers based Computers. on the SuperSymmetric symmetries.

Particles and Physics Laws Become Bit Patterns The space of the lattice has been tacitly discussed as if it was The SuperSymmetric Continuum Quantum Computer contains a space coordinate space. Actually we shall see in a few sections that the of bits that can describe particles. It also contains bit patterns continuum limit of a Quantum Computer (where the lattice spacing representing the interactions of particles. Both the particles in the shrinks to zero) can be treated as a second-quantized Fermi field. The space and the lagrangian possess a common symmetry. The lagrangian natural interpretation of the lattice space of a Quantum Computer is a is the program and the space is the memory of the computer. Together “momentum” space. they form one integral whole. The SuperSymmetric Continuum Quantum Computer adds bits So the particles’ momentum values and internal quantum numbers are representing the internal symmetries to each space-time point just as the data resident in the Quantum Computer. The momentum the Standard Model implements internal symmetries directly without distribution of a particle can be Fourier-transformed to a coordinate “deriving” them from a more fundamental concept such as a space- space representation. time with many extra dimensions beyond the 4-dimensional space- time of our experience that collapses to 4-dimensional space-time Reality is Reduced to Language generating the internal symmetries through a Kaluza-Klein Having seen particles first become symbols and then transform to bit mechanism. So we can view the SuperSymmetric Continuum patterns in a Continuum Quantum Computer™, having seen the Quantum Computer as the limit of a space-time lattice with a spin at interactions between bit patterns similarly reduced to bit patterns we each intersection point and a dangling thread of bits: now come to the conclusion that the universe and its physical laws – Reality – is in essence is linguistic – Language.

The Meaning of the Standard Model Language The purposes of language include understanding and communication. A language also reveals something of the thinking patterns of the speakers of the language. Language not only expresses our thoughts but it also helps shape them. What can one make of the linguistic approach to the Standard Model? What does the language of the Standard Model tell us about Nature? What are its capabilities for expressing ideas? Where is the Rosetta stone that could map the Figure. Lattice representation of SuperSymmetric Continuum language of the Standard Model to a form that we can understand in Quantum Computer with bits for the internal symmetries. The terms of its purpose and its goals? What are the reasons for its form? Continuum Quantum Computer emerges as the limit when the lattice spacing decreases to zero. The elementary particles that appear in the Standard Model and its relatively simple form is amazing considering the possibility of a 34 universe with untold numbers of unrelated particles each described by A reasonable starting point is to consider the set of computer a different theory. Many physicists complain the Standard Model has languages with which we are familiar: COBOL, FORTRAN, C, C++, too many particles. But actually it is a small miracle that a theory Java, and so on. Given one of these languages could we deduce its containing a few particles (36+), and describable by a lagrangian with purpose, objectives and “philosophy”? Imagine yourself an alien and a few parts, accounts for all known phenomena involving matter and someone hands you a description of, say, the C++ language describing radiation in the universe (except gravity). the features of C++ programming (written in your alien language). Could you deduce the purpose and “philosophy” of this language? An This being so, one has to ask if there is a human-understandable interesting question. interpretation of the theory at a level that can be explained in a few words understandable to an informed layman. (Feynman once said any The C++ language is used for mathematical calculations in science, good theory should have a simple idea behind it.) engineering and other practical applications. It is also used to manage devices such as printers, other computer devices and industrial John A. Wheeler developed a similar question in one of his papers10: machinery. C++ implements a “philosophy” called Object Oriented “An unfamiliar computer from far away stands at the center of the Programming. Could we have deduced this “philosophy” from the exhibition hall. Some of the onlookers marvel at its unprecedented structure of the C++ language? Personally, I think not – not without power; others gather in animated knots trying, but so far in vain, to serious effort – and not without an understanding of the drawbacks of make out its philosophy, its logic, and its architecture. The central idea the language that preceded it – the C programming language. of the new device escapes them. The central idea of the universe escapes us. [italics added]” What is the central idea behind the The C language was excellent for small scale programming projects. language of the Standard Model Quantum Computer and thus the But because it required the programmer to track every variable and all material universe? Have we not found Wheeler’s “unfamiliar computer the details of a program the C language was not appropriate for large from far away” to be our universe as described by the Standard programming projects. The C++ language was developed to handle Model? large-scale programming. It implemented a design philosophy called Object Oriented Programming. This approach was based on grouping Computer Languages and the Standard Model Language variables into objects. Objects were used as the central entities in One way to begin to develop an understanding of the central idea of programming and they were manipulated in C++ programs. By the universe is to first explore the more limited possibility of deducing grouping many variables within an object and then manipulating the the purpose of a computer language from its features. We wish to take objects C++ programs are easier to create, and easier to maintain and a language and deduce its purpose and goals from the structure and modify. capabilities of the language. Then we would like to apply this approach to the Standard Model computer language to develop a A casual examination of C++ language features would not reveal these deeper understanding of the universe. concepts and purposes. A detailed study of the C++ language might lead an imaginative alien to the rationale for the object-oriented features of C++.

10 J. A. Wheeler, The Computer and the Universe, International Journal of Returning to physics, the example of the C++ language suggests that Theoretical Physics, 21, 557 (1982). we might understand the language of the Standard Model and its 35 “philosophy” by understanding it within the context of other Each 2-state observable ni or mi can have the values 0 or 1 (bit “off” or languages, and the difficulties with other languages that the Standard bit “on”). This approach is equivalent to the spin ½ picture where the Model language resolves. This suggestion argues for a meta-theory of values are –½ or +½. physics theories: a theory of the nature of physics theories and of their comparative features – a Theory of Theories as we discussed earlier. The basis states can be easily interpreted as a set of multi-particle states in which each 2-state observable corresponds to a quantum level Equivalence of Continuum Quantum Computers and that can contain zero particles or one particle.12 Thus we can view the Second-Quantized Fermi Fields – Fock Space basis of Quantum Computer states as a number representation for The Continuum Quantum Computer™ can also be viewed as defining fermions (combined with the tape head state). a Fock space that leads to a second-quantized Fermi field representation. The formalism of second-quantized Fermi fields can be In the discussion that follows we will begin by lumping the 2-state used to describe Quantum Computers in the continuum limit. They can observables of the processor together with those of the computer tape also approximate Quantum Computers defined by a discrete lattice. for the sake of simplicity. (We will ignore the tape head for the moment.) The discussion will then treat these sets of observables A Quantum Computer can be described11 as a processor consisting of a separately. set of M 2-state (up or down) observables denoted {ni} where i ranges from 0 through M – 1, and an infinite memory consisting of a set of 2- The vacuum state for the number representation is the vector with all state observables denoted {mi} where i ranges from -∞ to ∞. (∞ is the 2-state observable values 0: symbol for infinity.) The computer’s memory can be viewed as either linear like a tape or arranged in a lattice that is numbered linearly. The V = | 0, 0, 0, …, 0; … 0, 0, 0, … > tape head is assumed to be “reading” at a memory location k on the tape. or, in short,

The state of a Quantum Computer can be specified with a basis of state V = | 0; 0 > vectors of the form: Creation and destruction operators can be defined that create and @ | n0, n1, n2, …, nM-1; … m-1, m0, m1, … >| k> connect multi-particle states (states with one-bits). The operator di creates a particle in the ith quantum state or, in other words, changes or, in shorthand notation, the ith 2-state observable bit from zero to one.

| n; m >| k> ith quantum state (2-state observable) The vector | k> specifies the state of the tape head. These vectors are o @ eigenvectors of the observables and span the space of states of the di V = | 0, 0, …, 1, … 0, 0, 0, … > Quantum Computer.

12 J. D. Bjorken and S. Drell, Relativistic Quantum Fields, (McGraw-Hill Book 11 D. Deutsch, Proceedings of the Royal Society of London, A 400 97 (1985). Company, New York, 1965) p. 46ff. 36 th The operator di destroys a particle in the i quantum state or, in other Pulling these considerations together we obtain the anticommutation words, changes the ith 2-state observable bit to zero from one. relations: Applying a destruction operator to the vacuum state annihilates it: @ { di, dj } = ij diV = 0 { di, dj} = 0 In order to guarantee the 2-state nature of the quantum levels @ @ @ (observables) we impose anti-commutation relations on di and di. { di , dj } = 0

@ @ @ { di, di } = didi + di di = 1 where ij equals 1 if i = j and ij equals zero otherwise.

{ di, di} = 0 The general Quantum Computer state can now be represented by a multi-particle state: @ @ { di , di } = 0 @ @ @ @ | n0, n1, …, nM-1; … m-1, m0, m1, … >| k> = N di dj … dq dp v|k> These anticommutation relations guarantee the spectrum of each observable will be one or zero (See Bjorken and Drell for the details). where N is a normalization constant. While it would be possible to go The relation of di and dj where i and j are different is not as clear until on and discuss normalizing states and other issues of the number we realize that the states representation we will refer the reader to Bjorken and Drell for further details.

@ @ @ @ di dj V = cdj di V The aspect that we would like to explore further is the introduction of a second-quantized Fermi field to characterize a Quantum Computer. for i ! j where c is a phase factor (ei&). The order in which bits are set For this purpose we define the field cannot influence the state that is created except for an irrelevant phase factor. (It is irrelevant because probabilities are calculated from the )(z, t) = ui(z, t)di absolute value squared of amplitudes.) The phase factor can be chosen i to be –1 to obtain consistency with the anticommutation relations above. Similarly where the sum is over the lattice sites labeled with index number i. The Hermitean conjugate field is

di dj = c’dj di @ )*(z, t) = ui*(z, t)di i i& for i ! j where c’ is a phase factor (e ). The phase factor can again be The operators ) and )* satisfy the anti-commutation relations chosen to be –1 to obtain consistency with the anticommutation relations above. {)*(z, t), )*(z’, t)} = 0

37 {)(z, t), )(z’, t)} = 0 Computer the measurement of a quantum bit(s) at one point in a Quantum Computer and the measurement of another bit in another part {)(z, t), )*(z’, t)} = ui(z, t) ui*(z’, t) = (z – z’) of the computer that can be connected to it by a light signal are i incompatible – they satisfy an uncertainty principle that prevents their where the set of functions ui(z, t) form a complete orthonormal set of simultaneous measurement. functions. These functions should be solutions of an equation that describes the time evolution of the quantum computer (perhaps similar Having arrived at a second-quantized Fermi field description of a to the Schrodinger equation of Quantum Mechanics). They satisfy the Quantum Computer it is interesting to raise the question of how it can orthonormality condition: be programmed from the perspective of this representation. Programming a Quantum Computer requires the specification of a ¶dz ui(z, t) uj*(z, t) = ij quantum dynamics that describes the evolution of the Quantum Computer from an initial state. The parameter z can be treated above as a one-dimensional spatial parameter. There are several approaches that appear reasonable at first glance:

In the continuum limit the index i can be transformed into a continuous 1. Specify a multi-particle (multi-bit) Hamiltonian using a variable that can be viewed as the momentum conjugate to the z Schrodinger-like equation that describes the time evolution of the parameter. particles (bits) of the Quantum Computer. This approach is limited by the fact that the number of particles (bits) is fixed – unlike most Thus we have arrived at a second-quantized Fermi field )(z, t) computations where the number of particles (on bits) varies as the formulation that describes a Quantum Computer. computation progresses.

A one-particle state can be specified by: 2. Define a second-quantized field theory for ) in z space with th i interactions that specify a computation. The interactions can @ ¶dz ui(z, t) )*(z, t)V = di V = | 0, 0, … , 0, 1, 0, … > appear in a variety of forms. Renormalization questions (infinities that appear in computations) are not an issue since the discreteness Multi-particle states can be constructed through a straightforward of the Quantum computer (the lattice spacing) provides a natural generalization. cutoff eliminating problems with infinities. Interaction terms that could appear in a second-quantized field theory include: The non-zero anti-commutation relation between ) and )* reflects a measurability issue for Quantum Computers that is familiar from the analogous phenomena in the theory of Quantum Fields. If we attempt a1)*))*) + a2)*Ø))*Ø) + … to measure a field at two points that can be connected by a light signal Øz Øz then an uncertainty principle exists that prevents the exact 13 measurement of the field at each point. In the case of a Quantum This approach has the difficulty that it is difficult to map the interaction terms into bit operations in a direct way making it difficult 13 N. Bohr and L. Rosenfeld, Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd., 12, to specify a program for a Quantum Computer in this manner. 8 (1933) and Phys. Rev. 78, 794 (1959). 38 alternately a superposition of Quantum Computer states with one bit 3. Directly use the creation and annihilation operators to define a “on”). program for a Quantum Computer. A simple example of this approach is to consider a Quantum Computer that embodies the dynamics of an The preceding example illustrates a Quantum Computer programmed “on” bit “moving” on a tape. We assume a set of simple time- to evolve dynamically according to a specified Hamiltonian. More independent Hamiltonian terms implement the dynamics: realistic, and therefore more complex, programming can be done with this approach using creation and annihilation operators. @ H = di+1 di i The preceding example illustrates the case of a nearest neighbor interaction – an interaction between adjoining bits or tape locations. th Let us assume the Quantum Computer starts in a state with only the 0 The Quantum Computer state evolves into a state composed of a th bit on (or in other words a particle at the 0 lattice or tape position). superposition of one “on” bit (one-particle states) states. Let us also define the set of one particle states: We have just seen the mapping of a Quantum Computer to a many- @ i = d i V particle formalism and to a second-quantized Fermi field without differentiating between the processor and the computer memory and Then with the time evolution operator: neglecting the positioning of the tape head. These complications can easily be handled within the framework of the mapping. S = eiHt As pointed out earlier, the general state of a Quantum Computer can we obtain the state of the Quantum Computer (initially in state 0) at be symbolized by time t:

| n0, n1, n2, …, nM-1; … m-1, m0, m1, … >| k> (t) = S0 or in expanded form: We can define creation and annihilation operators for the memory bits mi as before: ∞ n @ n (t) = (it) ( dj+1 dj ) 0 @ { di, dj } = ij n=0 n! j

{ di, dj} = 0 ∞ = (it)n n { d @, d @} = 0 n=0 n! i j where equals 1 if i = j and equals zero otherwise. In addition we The state of the Quantum Computer has evolved into a superposition ij ij of one-particle states residing at memory positions 0, 1, 2, … (or now define creation and annihilation operators for the processor bits which we will denote ai having similar anticommutation relations:

39 @ { ai, aj } = ij where the sum is over the ) lattice sites labeled with index number i and for & over the & lattice sites labeled with index number j. The ) { ai, aj} = 0 and & lattices are of course different. The Hermitean conjugate fields are @ @ { ai , aj } = 0 @ )*(z, t) = ui*(z, t)di Lastly we define a creation operator c@ and an annihilation operator c i corresponding to the tape position observable. Assuming the tape is and @ semi-infinite – the tape locations range from 0 to + . Therefore the &*(z, t) = wj*(z, t)aj ∞ j spectrum of this observable – the tape read position – ranges from 0 to +∞, and the creation and annihilation operators should be defined to The operators ) and )*, and & and &*, satisfy the anti-commutation satisfy Bose commutation relations: relations [ c, c@] = 1 {)*(z, t), )*(z’, t)} = 0 [ c, c ]= 0 {)(z, t), )(z’, t)} = 0 [ c@, c@] = 0

{)(z, t), )*(z’, t)} = ui(z, t) ui*(z’, t) = (z – z’) With these additional operators we can now define a Quantum i Computer state as: {&*(z, t), &*(z’, t)} = 0 | n0, n1, …, nM-1; … m-1, m0, m1, … >| k> =

@ k @ @ @ @ @ @ @ @ {& (z, t), & (z’, t)} = 0 N (c ) ar as … at au di dj … dq dp V

{& (z, t), &*(z’, t)} = wjz, t) wj(z’, t) = (z – z’) where V is the vacuum state of the Quantum Computer and N is a j normalization constant. where the set of functions ui(z, t) and wj(z, t) form complete Following the same strategy as earlier it is possible to define fields orthonormal sets of functions. In the case of the processor bits (2-state observables or particles) the number of observables is assumed to be infinite (although the number of occupied states is finite.) These )(z, t) = ui(z, t) di functions ) and & should be solutions of equations that describe the i time evolution of the quantum computer processor and tape. Normally and one would expect the processor and tape states to be interconnected as & (z, t) = wj(z, t) aj they evolve with time. j

40 The parameter z has been treated above as a one-dimensional spatial development of SuperStrings. This possibility will be considered in a parameter. It could have been a two-dimensional, three-dimensional, few chapters. etc. parameter. We could have started with a multi-dimensional tape or a set of multidimensional tapes – a Polycephalic Quantum Purpose of the Standard Model Language Computer™. The question still remains: What is the purpose of the language of the Standard Model if viewed as a computer language? Isaac Asimov The tape head observable with creation and annihilation operators c wrote a science fiction story that suggests one possible answer to this and c@ should not be mapped to a corresponding field since there is question. In Asimov’s story a very powerful computer is developed only one observable in the case presented. This situation can however and expanded until it meets all the needs of people and can answer be changed by introducing an array (lattice) of tape heads. Classical practically any question. One day a child asks its parent “What is the polycephalic Turing machines with multiple tape heads have been purpose of the universe?” The parent says we will ask the all-knowing discussed in the literature. These Turing machines had a finite number computer. The computer replies “Insufficient data but I will work on of tape heads. We can extend the number of tape heads to be infinite in it.” The computer grows and grows until space is not big enough for it number (and perhaps form a multidimensional lattice) in the case of so it moves itself to SuperSpace and eventually fills it up as well. Quantum Computers. This extension enables us to introduce a second- Eventually long after the race of its creators has disappeared and it is quantized Bose field corresponding to the lattice of tape heads: the only thing remaining in existence the computer finds the answer and says “Let there be Light.” "(z, t) = sj(z, t)cj j Is Asimov’s story an answer? Does the language of the Standard Model relate to the fundamental purpose of the universe? Is the @ "*(z, t) = sj*(z, t)cj universe a computation? For Whom? j

The Bose operators " and "* satisfy the commutation relations

["*(z, t), "*(z’, t)] = 0

[" (z, t), " (z’, t)] = 0

[" (z, t), "*(z’, t)] = sj(z, t) sj*(z’, t) = (z – z’) j

The appearance of both Fermi and Bose fields in Quantum Computers, and the apparent similarity of the lattice space of the Quantum Computer memory to a Supersymmetric space that could support Supersymmetry, leads us to consider the possibility that a Quantum Computer formalism may provide a foundation for an ad hoc 41 “Well you have 37 dimensions (degrees of freedom) to play around with in SuperString theory – a 26-dimensional space, a 10-dimensional space and a 1-dimensional space (“along the string”) in the currently most acceptable SuperString theory built around heterotic strings. Thirty-seven dimensions versus thirty-seven particles (interesting coincidence). You create 88 particles from space-time vibrations. We assume particles are fundamental. Aren’t 37-dimensions just as bothersome as 37+ fundamental particles (counting the graviton)? Too many dimensions! You have transformed the multitude of particles into a multitude of dimensions. It remains to be seen whether AA SSuuppeerrSSttrriinngg SuperString theory is actually a deeper theory, or perhaps the Ptolemaic theory of particles.

QQuuaannttuumm TTuurriinngg The Standard Model is experimentally verified. There is no convincing evidence for SuperStrings or for the hidden MMaacchhiinnee collapsed dimensions of SuperString theory.” The SuperString theorist replies

Introduction “We have reduced matter to artifacts of space-time and built In this chapter we will see how SuperString Theory stacks up as the theories that unite gravitation with the other forces of nature in successor to the Standard Model. In an earlier chapter we explored the a consistent way. We are within reach of a grand unified theory idea that the Standard Model has a representation as a language with a of all natural phenomena based on space-time geometry – the grammar specified by the Lagrangian of the theory and with the Holy Grail of Physics. The extra dimensions we introduced particles playing the role of the alphabet. may number 37 – we really count them as 26 with 10 other dimensions mapped to 10 of the 26 dimensions (although we The questions that we now address are “Is there a better formulation of can’t explain how or why as yet). But the rotational symmetry the theory of elementary particles than the Standard Model?” and “Are of this 26-dimensional space closely links the dimensions particles themselves reducible to something simpler?” without together so we don’t have a lot of arbitrary constants in our reference to the formulation in the previous chapter. theories unlike the Standard Model which has many arbitrary constants in its formulation.” Is SuperString Theory Simpler than the Standard Model? SuperString theorists say the Standard Model can’t be right because The Standard Model theorist responds there are too many “so-called” fundamental particles – at least 37 particles – and because it doesn’t include the force of gravity. Standard “If you figure out how the dimensions collapse to the four Model theorists can look at SuperString theory and say, space-time dimensions of the real world, then isn’t it possible that the collapse may involve the introduction of arbitrary or 42 incalculable constants as we see in the Standard Model? representation of a particle) or as a piece of cotton thread with two Arbitrary constants can appear in a spontaneous collapse such ends (open string representation of a particle). as you envision for the collapse of the dimensions.” Particles in the Standard Model are point-like. SuperString Theory The SuperString theorist lapses into poetry and has the last word: treats each “fundamental” particle – quarks, electrons, and so on – as an extended string in space-time. The strings are so small that their “We will continue to strive and seek and not to yield until we string-like nature cannot be verified experimentally with today’s find the correct formulation of the SuperString theory of the experiments. Current experiments see particles as point-like. If universe.” experiments could probe more closely they might see particles as strings. Strings are thought to be approximately 100 billion billion The Ideas of SuperString Theory times smaller than a proton. The best current attempt to develop a deeper theory than the Standard Model is called SuperString Theory. This theory attempts to explain Some versions of String Theory assume a particle is a closed string. the nature of elementary particles (What is a letter?) and how This type of string is often implemented in a form called a heterotic elementary particles interact with each other (What is the grammar?) string. A heterotic string consists of two strands (see previous chapter based on a deeper picture consisting of strings vibrating in a space for an example). One strand is a 10-dimensional string. The other with many more dimensions than the four space-time dimensions with strand is a 26-dimensional string. This composite string has properties which we are familiar. that appear to be closer to reality than other types of strings.

The linguistic representation of the Standard Model described in an In SuperString Theories elementary particles are vibrations of the earlier chapter is an interesting preliminary to the SuperString Theory. strings. A string can vibrate in a number of different ways just like a We will look at SuperString Theory from a Quantum Grammar™ and violin string can vibrate at different frequencies. Each vibration of a Quantum Turing Machine perspective in this chapter. SuperString corresponds to a different particle. SuperString theory should have a vibration for each different type of elementary particle There is no direct connection between the strings of particles (which including gravitons – the particle that carries the force of gravity. are actually words made of characters) in our linguistic representation Actually SuperString theories predict many more particles than the of the Standard Model and the SuperStrings described in this chapter. particles that we have actually found in nature. Eliminating these extra The one thing these approaches have in common is that they are both particles – often by suggesting that they are too massive to be observed one-dimensional. A string of characters is one-dimensional – the by current particle accelerators – is a major task of SuperString theory. characters are lined up in a row. A string in SuperString Theory is one- dimensional – it is a mathematical construct that is analogous to a In 2500 years we have moved from the music of the spheres to the string made of thread. music of the strings: from divine harmony to the quantum string.

A string in the linguistic representation of the Standard Model is a discrete list of individual particle characters. A string in String Theory is a continuous, one-dimensional space (like a thread) representing one particle. It can be visualized as a rubber band (closed string 43 SuperString theories have many interesting features that are described Having seen the basic ideas of SuperString theories we now try to in popular14 and technical15 books. Perhaps the most interesting feature relate them to the linguistic representation of the Standard Model. Is of SuperString theories is their high degree of symmetry. First they SuperString theory interpretable as a computer language? have the rotational symmetry of a 26-dimension space-time. Then they have conformal symmetry – a symmetry that can crudely be described Equivalence of Standard Model & SuperString Theory? as stretchability – a string can be stretched or deformed in The particle representation of the Standard Model is one theoretical mathematical ways. They also have a peculiar symmetry called description of nature. The SuperString theory may be an alternate Supersymmetry. Supersymmetry is based on the rotation of fermions description of nature based on vibrating strings in a higher- (spin ½ particles, spin 3/2 particles, and so on) and bosons (spin 0 dimensional space-time. particles, spin 1 particles, and so on) into each other. There are simple solvable models that have been discovered in recent Before Supersymmetry, fermions were fermions and had special years that suggest that these theories could eventually grow into properties that distinguished them. For example, they had Fermi-Dirac complementary representations. These simple models have a space- statistics based on the Pauli Exclusion Principle that stated that no two time formulation that contains miniature black holes and no matter. fermions could have exactly the same set of quantum numbers. This formulation can be transformed into another representation that transforms the black holes into what appear to be particles. There is a Before Supersymmetry, bosons were bosons and they had their own geometrical space-time formulation without matter – and an equivalent special features. Bosons had Bose statistics that stated that any number particle formulation. of bosons could have the same set of quantum numbers. These differences in statistics and other features gave bosons and fermions These equivalent formulations may be hints of an equivalence between radically different properties. a suitably extended Standard Model of particles and a SuperString Supersymmetry enables rotations to take place in a mathematical space theory. of bosons and fermions. Fermions and bosons can be rotated into each other. As a result of Supersymmetry fermions and bosons are If we think of computer languages that are equivalent to each other but interrelated and form families. This symmetry puts important differ greatly in their appearance, can we view the Standard Model restrictions on theories that support it. (suitably extended) and SuperString theory as potentially equivalent formulations? Particles are the fundamental symbols in the Standard At the moment no solid experimental evidence exists that supports the Model Quantum Grammar. Perhaps space-time geometrical objects are idea that Supersymmetry is a symmetry of Nature. However the the fundamental symbols in the Quantum Grammar of SuperString Supersymmetry concept is very appealing theoretically and it is an Theory. integral part of SuperString theories. Linguistic Model of SuperString Theory If we take the idea of a computer language description of Nature seriously, then we have to understand how the features of 14 Michio Kaku, Hyperspace, (Bantam, Doubleday, Dell Publishing, New York, Supersymmetry theory can be viewed as symbols of a computer 1994) among others. language grammar and what the grammar of this language is. 15 Joseph Polchinski, String Theory (Cambridge University Press, New York, 1998). D. Bailin and A. Love, Supersymmetric Gauge Field Theory and String Theory (Institute of Physics Publishing, Philadelphia, PA, 1994) among others. 44 We can view the Standard Model as a computer language and the computer language grammar rules. The grammar rules can be Supersymmetry theory as a possibly more fundamental (lower level) represented pictorially as closed string diagrams: computer language. Is the Supersymmetry language the machine language upon which the Standard Model language is based? In this regard one thinks of computer languages such as C++ that are translatable to a lower level language such as assembly language and then to an even lower level language machine language. The hallmark Two particles of these lower level languages is that they are increasingly simpler in One particle nature with fewer and fewer features.

The lowest level computer language, machine language, has few Figure. Two visualizations of rule 1 for the “fission” of a string into constructs in it and directly tells the computer chips what to do. On the two strings. other hand, the interpretation of the constructs of a higher level language in terms of lower level language constructs is often complex. The “fission” of a photon string into an electron string and a positron string is an example of this grammar rule. The strings look like tubes A Language and Turing Machine for SuperString Theory or pipes as they move in time: if you take many pictures of a moving The natural choice for a grammar for a SuperString Theory has letters string and superimpose the pictures the motion of the strings will trace (terminal symbols) that are SuperStrings. Each letter represents a out tubes. That is why the visualizations appear as tubes. specific string vibration. The production rules for the SuperString grammar are simple at the symbolic language level. The details of the numeric calculations behind the grammar rules are complex. They are described in books such as those listed in reference 26.

A set of grammar rules for the linguistic representation of SuperString interactions is:

Rule 1. →

Rule 2. →

Rule 3. → ’ ’

The quote marks on the right side of rule 3 indicate that the symbols on the right side represent different particles (terminal symbols) from the particles on the left side. This notation is part of the formalism of

45 different. The SQTM generates intermediate string states based on the ’ details of the SuperString theory that it implements. An example of an interaction between strings with interesting intermediate states is:

→ ’’ → ’’’’ → ’’’ → ’ ’

The intermediate strings are generated using the three SuperString grammar rules stated earlier. This set of transitions can also be depicted with a SuperString theory diagram with time orderings:

’

Figure. A diagrammatic visualization of rule 3. The strings look like tubes or pipes as they move in time. In this case two strings interact and then produce two output strings.

Notice the grammar production rules for SuperStrings are much simpler than the grammar production rules for the Standard Model. This was to be expected if the SuperString Theory is in some sense a deeper theory corresponding to a lower level computer language – Figure. String diagram of two interacting strings that generate two perhaps the equivalent of machine language for particles. strings in intermediate states. These generated strings are absorbed so that only two strings emerge in the output. The intermediate state The Quantum Computer that corresponds to SuperString theory “words” are indicated by vertical dashed lines. Many other diagrams accepts input states consisting of a word containing a string of particle contribute to this string-string interaction when realistic calculations strings. It processes the input and generates an output word consisting are made. of a symbol string consisting of particle strings. Each possible output word has a corresponding probability of being produced since the The SQTM offers a different view of the features of SuperString Quantum Computer is probabilistic. Inside the Quantum Computer the Theory. This view may be helpful in deepening our understanding of input particles interact – producing a variety of intermediate particle the implications of SuperString Theory and its relation to the Standard states or words. Eventually the output particles are generated. Model.

The general features of the SuperString Quantum Turing Machine (SQTM) are similar to the Quantum Turing Machine for the Standard Model. However the language and the theory behind the language are 46 Beyond SuperStrings – Hidden Dimensions as Quantum Computer Tapes SuperString theories assume the existence of hidden dimensions that are the origin of the internal symmetries of elementary particles. These dimensions are somehow compactified (curled up) in a manner that is Internal not well understood. As a result we only experience the normal four Tape Space dimensions of space and time. Input output

The Quantum Computer representations of the Standard Model and Supersymmetry that we have developed suggest an alternate Tape tape possibility: the internal tapes of the Quantum Computer may define a different space from the space of the input and output particles.

A Quantum Computer has several tapes. These tapes play the role of computer memory just like computer tapes. One tape – the input tape – Figure. A Quantum Computer with an internal multidimensional space has the input particles on it. Another tape – the output tape – has the (tape). output particles on it. (The input and output tapes could be the same tape.) The internal Turing machine tape defines an internal “space” to which we can assign dimensions. This space, as it is normally viewed in The Quantum Computer also has an internal tape that can be viewed as Turing machines, is divided into cells. However in Quantum a set of tapes or as a multidimensional tape upon which intermediate Computers it can be made into a continuous space. Further we can states of the particles can be stored. We can view a Quantum make this space the space of the internal symmetries of the Standard Computer that processes an input state to produce an output state as: Model or Supersymmetry Theory (or in part the space of symmetries of the Standard Model or Supersymmetry Theory).

This concept leads us to view the space of internal symmetries as part of the Quantum Computer. This space is relevant when particles interact (collide). The internal space is not relevant and is not “visible” when particles are not interacting. This simple view agrees with our experience. The “space” of the input and output tapes interfaces with (or melds into) the internal space when particles interact. So we can expect that the internal space somehow meshes the input and output space in a smooth way.

The smoothness (continuity) of the transition between the internal space of the Quantum Computer and the external space of the input and output states could result in a unified theory that unites gravity and 47 the internal particle symmetries. This approach is similar in flavor to the standard SuperString approach for the unification of the forces of nature.

An interesting point related to the Quantum Computer representation of the Standard Model and Supersymmetry is quark confinement. Individual, isolated quarks have not been found in Nature. Quarks always appear bound together with other quarks: three quarks in a proton, a quark-antiquark pair in a pion, and so on. The inability to split a proton or any other particle containing quarks into individual quarks is an interesting phenomenon. Most theorists believe it is a feature of the Strong Interaction between quarks. Unlike other forces such as gravity the Strong Interaction becomes stronger and stronger as quarks move further apart preventing them from separating. This feature of the Strong Interaction can be restated in the grammar of the Standard Model or Supersymmetry by saying that quarks are nonterminal symbols.

A non-terminal symbol is a symbol that can appear within the grammar of a language but cannot be an input symbol or an output symbol. It only appears as an intermediate symbol within the grammar rules. Protons, pions and other composite particles composed of quarks can be viewed as terminal symbols – symbols that can appear as input symbols or output symbols. Quarks only appear inside the Quantum Computer as particles in intermediate states generated by the grammar. They are reassembled into composite particles that are output (terminal) symbols.

The Quantum Computer representations of the Standard Model and Supersymmetry provide an alternate view of elementary particles that is both simple and yet capable of embodying significant features of these theories in a straightforward way. Quark confinement is one Standard Model feature that is easily represented in the linguistic representation. The existence of hidden dimensions is also easily represented in the Quantum Computer approach as an internal multidimensional tape. The linguistic approach to elementary particle theories naturally accommodates major features of elementary particles. 48 solidity of our world of ordinary perceptions. A world governed by a language and consisting of nothingness. Matter being mere wrinkles in space and time.

Actually space and time themselves – since they are represented by the gravitational field which itself is a SuperString construct – also 99 become constructs. In the end we wind up with “nothing” – a “nothing” that develops structure through the SuperString Theory. The structure “becomes real” through the existence of SuperStrings – particles represented by mathematical entities that resemble open and closed strings. “Normal” space and time now exist through the QQuuaannttuumm presence of gravitons – particles carrying the gravitational force – which are in reality mathematical strings.

CCoommppuutteerrss aass tthhee In the previous chapter we saw the appearance of both Fermi and Bose fields in Quantum Computers. We also saw the apparent similarity of the lattice space of the Quantum Computer memory to a space that FFoouunnddaattiioonn ooff could support Supersymmetry. These observations lead us to consider the possibility that a Quantum Computer formalism may provide a SSuuppeerrSSttrriinnggss foundation for SuperString Theory. SuperString theory today has some notable successes. However it is still a patchwork quilt of clever ideas that lead to something Introduction approaching a realistic theory of elementary particles. The situation is Many physicists feel that SuperString Theory is the next step beyond reminiscent of the Bohr model of the atom. It blended together the Standard Model. This theory is based on an interesting idea: build disparate ideas (that could be viewed as incompatible in their origin) to matter out of the fabric of space and time. Not the space and time with produce a model of the Hydrogen atom that explained some major which we are familiar but rather a space and time of many more experimental features of Hydrogen. It was not until Quantum dimensions. Most of the dimensions of this higher dimensional Mechanics developed about ten years later that a logically satisfactory universe curl up into a “tight ball” so we only see the four dimensions theory of Hydrogen was created. of space and time of everyday experience. The curled extra dimensions are thought to be the origin of the internal symmetries that particles SuperString theory may be a “Bohr model” of elementary particles in possess in the Standard Model. wait for the right theory. The SuperString Quantum Computer that will be developed in this chapter offers a totally new view of the origins of The reduction of matter to a feature of space-time is an important SuperString theory. concept. It leads us to an insubstantial universe consisting only of space and time that is based on quantum probabilities. It lacks the 49 The Bosonic String Part of a Quantum Computer location (or 2-state observable), and Bose creation and annihilation The beginning point is the standard view of a Quantum Computer operators with each tape head. augmented to contain a set of tapes for memory and a set of linear one- dimensional arrays of tape heads. Each array has an infinite number of In the preceding chapter we defined a string of tape heads with tape heads and looks like a “tape” (or string) of tape heads. corresponding operators cn with n ranging from –∞ to ∞. We can redefine these operators16 to show they can eventually be used to develop a SuperString formalism. Let us define

an = cn n m 0 @ @ an = cn n m 0

a-n = cn n < 0 Processor @ @ … a-n = cn n < 0

Tape Heads The operators an and an and their hermitean conjugates can be generalized to multiple strings of tape heads by adding an index specifying the tape head string.

Bose operators for D strings of tape heads are:

an and an

with n ranging from 0 to ∞. The index specifies the tape head string and ranges from 0 to D.

Memory Tapes

Figure. A Polycephalic Quantum Computer™ with infinite arrays of tape heads and many memory tapes.

We will call an infinite linear array of tape heads a string of tape heads.

16 Following a similar path to the discussion in the preceding chapter we We attempt to use a notation consistent with standard texts on SuperStrings: D. Bailin and A. Love, Supersymmetric Gauge Field Theory and String Theory associate Fermi creation and annihilation operators with each tape (Institute of Physics Publishing, Philadelphia, PA, 1994) page 157; Joseph Polchinski, String Theory (Cambridge University Press, New York, 1998) 50 -∞ 0 +∞ These operators are the oscillator coefficient operators in the mode expansion of the closed bosonic string:

a 2 -2in($ - ") -2in($ + ") n X = x + l p $+ .5l ( n e + n e )/n an n ! 0 Notice we have effectively created a field just as we did in the th Figure. The string of tape heads. There is a tape head on the string previous chapter – although with the complications of an extra index. for each value of n with corresponding operators an for n m 0 and a-n The coordinate X is a coordinate in a D dimensional space. The for n < 0. parameters $ and " can be viewed as parametrizing a closed string in D dimensional space with a time-like coordinate $ and a space-like These operators have the same tape head commutation relations seen coordinate ". For fixed $, X traces out a curve (a string) as " varies. in the previous chapter. Taking account of the added index the The terms in the summation are for a “right mover” part that depends commutation relations become: on $ - " and uses the operators n , and for a “left mover” part that depends on $ + " and uses the operators n . The left mover modes are @ [ ai , aj ] = -ij used to construct heterotic strings – the type of string that appears to result in a theory that is closest to reality. [ ai , aj ] = 0 The mode expansion of X can be viewed as the solution of the one- @ @ [ ai , aj ] = 0 dimensional wave equation:

where is non-zero if g and has absolute value 1 if = . The 2 2 Ø - Ø X = 0 tape head operators for different tape head strings commute with each 2 2 @ Ø$ Ø" other. The operators ai and aj have similar commutation relations. Introducing a metric h ( , ) in addition to the metric = diag(1,-1) To establish a connection with the SuperString literature and to make a $ " that appeared earlier in the commutation relations we can develop the connection with the bosonic strings of Physics we introduce the mode expansion for X from the action for the relativistic bosonic operators n and n with n ranging from –∞ to ∞. The index again string specifies the tape head string. These operators are related to an and an via: ½ S = -T¶d$¶d" (- det h) h ØXØX @ n = „n an -n = „n an for n > 0 2

and We do this by obtaining the Euler-Lagrange equations of motion, and using reparametrization invariance and conformal invariance to reduce @ n = „n a n -n = „n an for n > 0 the metric h to 51 where ij equals 1 if i = j and ij equals zero otherwise. The indices i

h = and j are integers that label tape positions and range between -∞ and +∞. which then leads to the above wave equation. Let us now consider the Polycephalic Quantum Computer™ pictured The other features of bosonic strings (such as the Virasoro algebra) at the beginning of this chapter that has a number of memory tapes. can be obtained by consistently applying the standard field theoretic Let us suppose there are D tapes that are labeled with the index . formalism (reference 26). In order to make our discussion similar in form to the Physics As a result of this development we can see the tape heads of a literature (reference 26 and references therein) we now define Quantum Computer supplemented by reparametrization and conformal operators for Ramond boundary conditions: invariance can be seen as providing a representation of the bosonic string concept. Consistency requirements select the dimension D of the dn = bn n m 0 @ bosonic string space to be 26. Thus there are 26 infinite “strings” of d-n = bn n m 0 tape heads in the Quantum Computer for SuperString theory.

d-n = bn-1 n [ 0 @ This result provides a new concrete representation of bosonic strings. d-n = bn-1 n [ 0 In the next section we will see the memory tapes of a Quantum Computer can be mapped to SuperStrings. These new operators satisfy

The SuperString Part of a Quantum Computer { di, dj} = i+j,0 In the preceding section we saw how arrays (strings) of tape heads { di, dj} = i+j,0 could be related to bosonic strings. In this section we show the Fermi operators of memory tapes can be related to SuperStrings. In the next It is easy to generalize the above anticommutation relations to: section we join the bosonic string discussion together with the results of this section to show how heterotic strings can naturally emerge from { di , dj } = -i+j,0 the parts of a Quantum Computer. { di , dj } = -i+j,0

The preceding chapter showed that each tape position could be where i and j range from – to . @ ∞ ∞ associated with creation and annihilation operators bi and bj satisfying the anticommutation relations:

@ { bi, bj } = ij

{ bi, bj} = 0

@ @ { bi , bj } = 0

52 -∞ 0 +∞ and anti-periodic (Neveu-Schwartz) boundary conditions:

-2in($ + ") L = bn e n Z + 1\2 dn dn where the sum extends from –∞ to ∞. These mode expansions satisfy th Figure. The memory tape string. There is a location on the string the equations: for each value of n with corresponding operators dn for n m 0 and d-n for n < 0. Ø + Ø R = 0 Ø$ Ø" We can now form the mode expansions for a fermionic right mover, closed SuperString for periodic (Ramond) boundary conditions - = 0 (reference 26): Ø Ø L Ø$ Ø"

-2in($ - ") R = dn e and the canonical anti-commutation relations: n Z

% % and for anti-periodic (Neveu-Schwartz) boundary conditions: {R ($, "), R ($, "’)} = {L ($, "), L ($, "’)}

-2in($ - ") = -2 (" - "’) R = bn e n Z + 1\2 % {R ($, "), L ($, "’)} = 0 where the sum extends from –∞ to ∞ and where the operators bm are the corresponding Neveu-Schwartz operators with m half-integral. If we form the spinor

The operators bm are not the operators bn defined on the previous R ($, ") page. The operators bn and bn satisfy similar anticommutation ($, ") = relations to dn and dn (see below). They are an alternate set of L ($, ") creation and annihilation operators for the memory tapes labeled with half integer values. then we can define an action that leads to the above differential equations through the canonical approach from Euler-Lagrange We can now form the mode expansions for a fermionic left mover, equations of motion in the covariant gauge. The combined action for closed SuperString assuming periodic (Ramond) boundary conditions: SuperStrings (SuperSymmetry requires a bosonic string term) is

-2in($ + ") L = dn e n Z 53 2 ½ strings of tape heads with boson operators that support a 26- S0 = -1 d " (- det h) (h Ø X Ø X + i ! Ø ) ¶ dimensional space for bosonic strings. 2 A heterotic string is a closed string created by using right movers of a where (type II) SuperString and the left movers of a bosonic string. The space of states of heterotic strings is the direct product of the spaces of the 0 -i 0 i right and left movers. !0 = !1 = i 0 i 0 There is an apparent problem because the space of the left movers (bosonic strings) is 26 space-time dimensions while the space of the Consequently the entire formalism of SuperStrings can be “reverse right movers (SuperStrings) is 10 space-time dimensions. The engineered” from memory tapes in a Quantum Computer. resolution of the problem is to compactify 16 of the left mover dimensions by associating them with a 16-dimensional torus with Consistency requirements select the dimension D of the SuperString extremely small radii of the order of fundamental string length. So 16 space to be 10. Thus there are 10 infinite memory tapes, or memory dimensions shrink to almost nothing leaving 10 “real” dimensions for “strings”, in the Quantum Computer for SuperString theory. the left mover. These extra dimensions presumably play a useful role by generating the internal symmetries of the elementary particles (as The SuperString action also provides a 10-dimensional bosonic string found in the Standard Model) through the Kaluza-Klein mechanism. In sector. The memory of the Quantum Computer could be augmented to the Kaluza-Klein mechanism a higher dimensional theory containing support an additional 10 tapes with bosonic observables (not 2-state gravity generates Yang-Mills-like gauge fields when the number of observables) supporting a spectrum consisting of the positive integers. dimensions is reduced through compactification. This addition would provide consistency with the action.

The next section joins bosonic strings and fermionic Superstrings to create a heterotic string. Heterotic string theories appear to offer the closest connection to Reality as we currently know it.

The Quantum Computer as the Foundation of Heterotic Strings In the preceding two sections we have shown how to formulate a Quantum Computer equivalent to Superstring theory consisting of bosonic strings (from tape heads) and fermionic SuperStrings (from memory tapes). In this section we will see how to build a heterotic string within the framework of the SuperString Quantum Computer.

The SuperString Quantum Computer has ten infinite memory tapes with fermionic operators that support a 10-dimensional SuperString space with SuperStrings. It also has twenty-six infinite arrays or 54 i b -½|0>R being right mover SuperStrings of the Neveu-Schwartz type j Tape Heads and with -1|0>L being bosonic left mover strings.

The combined role of the tape heads and the memory tapes supports the creation of heterotic SuperString particle states.

… … Processor The next issue that the SuperString Quantum Computer must address is interactions between the SuperStrings. SuperString theory does not 10 Head Strings 16 Head Strings have an intrinsic well-defined way of specifying interactions between (Compactified) SuperStrings. So we cannot look for strong guidance from SuperString theory when we try to develop interactions within the framework of the SuperString Quantum Computer.

SuperString Theory does have a principle that is used to specify interactions – an isomorphism between states and vertex operators. This principle is a feature of Conformal Field Theories. SuperString theories are Conformal Field Theories.

The vertex operators of SuperString theories are used to define interactions between SuperStrings. A vertex operator is necessary whenever SuperStrings “intersect” – when a SuperString fissions into 10 Memory Tapes two SuperStrings or two SuperStrings amalgamate to form one SuperString. Figure. SuperString Quantum Computer with 26 infinite arrays (strings) of tape heads and 10 memory tapes. Since we know how to define SuperString states using the raising (creation) and lowering (annihilation) operators of the SuperString The creation and annihilation operators of the tape heads and memory Quantum Computer we can use that information and the isomorphism tapes can be used to create states corresponding to elementary (correspondence) to define vertex operators within the SuperString particles. For example, the combined tape head-memory location state Quantum Computer. In fact we can define states corresponding to a i j vertex operator in the SuperString Quantum Computer processor. b -½|0>R -1|0>L i, j = 1, … 8 In view of its role the SuperString Quantum Computer processor can be decomposed into a traceless symmetric 10-dimensional appears to be the natural place to specify SuperString interactions. The graviton, an anti-symmetric tensor and a scalar dilaton. The state is processor is supposed to contain the program of the computer and the defined using |0> as the vacuum state (empty state) for right movers R initial state of the computer – the input. and |0>L as the vacuum state (empty state) for left movers, with

55 An alternative to using the processor to specify interactions is to use a Quantum Computer formulation avoids this issue by basing standard vertex operator expressed in SuperString raising and lowering SuperString theory on computation. operators. An example is the normal ordered vertex operator: On the other hand, one could argue that a space-time approach should not be ruled out just because the distances are ultra-short. It works well ik•X :e : at the larger distances with which we are familiar. Perhaps viewing short distance physics as different because we can’t “see” at that level where k is the momentum of a SuperString particle and X is the is the subtle head of anthropomorphism rearing up. Why shouldn’t the SuperString coordinate operator. space-time approach work equally as well at the short distances that we cannot directly perceive as the larger distances that we can directly Is the SuperString Quantum Computer More Fundamental perceive? than Coordinate Space SuperString Theory? The view that we have espoused in this discussion is that the Well there is one example where the extrapolation from common foundation of SuperStrings is in Quantum Computers. The raising and experience to shorter distances has abjectly failed: quantum lowering operators are the key features of the formalism. If one looks phenomena. The theories of mechanics and electromagnetism worked at the development of SuperString theories (reference 15 and beautifully at the level of everyday experience. We needed a major references therein) then it is very evident that the detailed study of change – quantization - to develop a theory that worked at the sub- almost all SuperString features is based on raising (creation) and atomic short distance scale. Is the foundation of SuperString theory a lowering (annihilation) operator expressions. new level with Computational Quantization?

In the standard development of SuperString Theory raising and lowering operators are “derived” from a space-time formalism filled with nice symmetries. In our approach the raising and lowering operators of the SuperString Quantum Computer are the fundamental constructs and the space-time formalism is secondary.

SuperString Quantum Computer™ Space-Time Theory

Establishing a Quantum Computer foundation for SuperString Theory changes the perspective from a space-time formalism to a more fundamental computational formalism. We have freed SuperStrings from the trappings of space-time!

One might argue that space-time approaches to the development of physical theories have been the only successful approach in the past. But one might also argue that a space-time approach is meaningless at ultra-short distances where space and time itself are in question. The

56 Computer Machine and Assembly Languages The traditional (non-quantum) computer can be viewed simply as a main memory, an accumulator or register (modern computers have many registers), and a central processing unit (CPU) that executes a program (instructions) step by step. It can be visualized as: 1100 Memory Memory Address Word 0 CPU Word 1 Word 2 QQuuaannttuumm CCoommppuutteerr …3 PPrroocceessssoorr Register Figure. Simplified model of a normal computer.

OOppeerraattiioonnss aanndd A set of data and a program (or set of instructions) is stored in memory and the CPU executes the program step by step using the data to QQuuaannttuumm CCoommppuutteerr produce an output set of data. The basic instructions of assembly language and machine language move data values between memory and the register (or registers), LLaanngguuaaggeess manipulate the data value in the register and provide basic arithmetic and logical operations17:

Introduction LOAD M – load the value at memory location M into the A natural question that arises when one considers Quantum Computers register is the role of the Quantum Computer processor and the operations it supports. A further question of some interest is whether a quantum STORE M – store the value in the register at memory location machine language exists and what its nature might be. Lastly the M question of higher level languages is also relevant. Can we develop a Quantum Assembly Language™? What is the nature of High Level SHIFT k – shift the value in the register by k bits Quantum Languages™? Are there, for example, equivalents to the C or C++ languages?

17 See for example Kurt Maly and Allen R. Hanson, Fundamentals of the Computing Sciences (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1978) Chap. 8. 57 The following arithmetic instructions modify the value in the register. TRA M – set the value of the program counter to the value at The AND, OR and NOT instructions perform bit-wise and, or and not memory location M operations. TZR M – set the value of the program counter to the value at ADD M – add the value at memory location M to the value memory location M if the value in in the register the register is zero.

SUBTRACT M – subtract the value at memory location M HALT – stop execution of the program from the value in the register The above set of instructions form an extremely simple assembly MULTIPLY M – multiply the value in the register by the language. They also are in a one-to-one correspondence with machine value at memory location M instructions (machine language). Most assembly and machine languages have a much more extensive set of instructions. DIVIDE M – divide the value in the register by the value at memory location M Algebraic Representation of Assembly Languages The normal view of assembly language is that it has a word or AND M – change the value in the register by “anding” it instruction oriented format. Some assembly language programmers with the value at memory location M would even say that assembly language is somewhat English-like in part. OR M – change the value in the register by “oring” it with the value at memory location M Computer languages in general have tended to become more English- like in recent years in an attempt to make them easier for NOT – change the value in the register by “not-ing” it programmers. Some view a form of highly structured English to be a goal for computer programming languages. The following instructions implement input and output of data values. In this section we follow the opposite course and show that computer INPUT M – input a value storing it at memory location M languages can be reduced to an algebraic representation. By algebraic we mean that the computer language can be represented with operator OUTPUT M – output the value at memory location M expressions using operators that have an algebra similar to that of the raising and lowering operators seen earlier. We will develop the A computer has another register called the Program Counter. The algebraic representation for the case of the simple assembly language value in the program counter is the memory location of the next described in the previous section. There are a number of reasons why instruction to execute. The following instructions support non- this reduction is interesting: sequential flow of control in a program. A program can “leap” from one instruction in a program to another instruction many steps away 1. It may help to understand SuperString dynamics more and resume normal sequential execution of instructions. deeply (later in this chapter).

2. It will deepen our understanding of computer languages. 58 3. It provides a basis for the understanding of Quantum Register Computers. Word - infinite … 4. It may have a role in research on one of the major questions CPU of computer science: proving a program actually does what Program Counter it is designed to do. Algebraic formalisms are generally easier to prove theorems then English-like formalisms. Word - infinite …

The algebraic representation can be defined at the level of individual Address Memory bits based on anti-commuting Fermi operators. But it seems more 0 Word - infinite … appropriate to develop a representation for “words” consisting of some 1 Word - infinite … number of bits. An algebraic representation for a word-based assembly 2 $ $ $ language can be developed using commuting harmonic oscillator-like . raising and lowering operators. . . A word consists of a number of bits. In currently popular computers the word size is 32 bits (32-bit computer). The size of the word determines the largest and smallest integer that can be stored in the word. The largest integer that can be stored in a 32-bit word is 4,294,967,294 and the smallest integer that can be stored in a 32-bit word is 0 if we treat words as holding unsigned integers. Figure. Visualization of a Computer with infinite words. To establish the algebraic representation we associate a harmonic To develop a simple algebraic representation of assembly language we @ will assume the size of a word is so large that it can be viewed as raising operator ai and a lowering operator ai with each memory infinite to a good approximation. (It is also possible to develop location. These operators satisfy the commutation relations: algebraic representations for finite word sizes.) As a result memory @ locations can contain non-negative integers of arbitrarily large value. [ ai, aj ] = ij [ ai, aj] = 0 @ @ [ ai , aj ] = 0

where ij is 1 if i = j and zero otherwise. We define a pair of raising and lowering operators for the register r and r@ with commutation relations

@ [ ri, rj ] = ij [ ri, rj] = 0 59 @ @ [ ri , rj ] = 0 Set the Value at Memory Location m to Zero Nm (am) The ground state of the computer is the state with the values at all Mm = memory locations set to zero. It is represented by the vector ªNm!

The above expression for M is symbolic. The expression represents | 0, 0, 0, … > h | 0 > h V m the following expression in which the operators are carefully ordered A state of the computer will be represented by a vector of the form to avoid complications (c-numbers etc.) resulting from reordering.

q @ n @ m @ p q | n, m, p, … > = N (r ) (a0 ) (a1 ) … | 0 > (ln am) Nm 1 Mm h q where N is a normalization constant and with the first number being q! ªNm! the value in the register, the second number the value at memory location 0, the third number the value at memory location 1, and so on. where the sum ranges from 0 to ∞. When Mm is applied to a state it For simplicity we will not consider superpositions of computer states sets the value of the mth memory location to zero. at this point. We will discuss superpositions later in this chapter. n Within this limitation we can set a computer state to have certain Mm | … , n, … > = (am) | … , n, … > initial values in memory and then have it evolve by executing a ªn! “program” to a final computer state with a different set of computer values in memory. The “program” is a mapping of the instructions of mth memory location value an assembly language program to algebraic expressions in the raising and lowering operators. = | … , 0, … >

Basic Operators of the Algebraic Representation The repeated application of factors of Nm to the state results in factors The key operators that are required for the algebraic representation are: of n.

Fetch the Value at a Memory Location (Number Operator) Change the Value at Memory Location m from 0 to the Value at @ Nm = am am Location n For example, n @ Nn (am ) Pm = Nm | … , n, … > = n | … , n, … > ªNn!

n The above expression for Pm is also symbolic. The expression mth memory location value represents the following expression in which the operators are carefully ordered to avoid complications (c-numbers etc.) resulting from reordering. 60 corresponding to 20. The next bit is bit 1 corresponding to 21 and so q n @ q on. (ln am ) Nn 1 Pm h q If the bit shift is to the right (k > 0) then we assume the padding bits q! ªNn! are 0’s. For example a shift of the bit pattern for 7 = 1110000 … one n bit to the right is 14 = 01110000 … As a result the value in the register where the sum ranges from 0 to ∞. When Pm is applied to a state it changes the value of the mth memory location from zero to the value at is doubled (k = 1), quadrupled (k = 2), and so on. The algebraic the nth memory location. expression for a k bit right shift is

q th th @ q m n (ln ar ) Sr ªNr!

q q! ªTr!

n @ x Pm | … , 0, … , x, … > = (am ) | … , 0, … , x, …> ªx! where k Sr = (2 – 1)Nr = | … , x, … , x, … > and The application of the factors of Nn to the state results in factors of x k that lead to the above expression when summed. Tr = 2 Nr

n The operators Mm and Pm enable us to simply express the algebraic If the bit shift is to the left (negative k), then we assume zero bits are equivalent of assembly language instructions: added “at ∞”. If k = -1 then the effect of left shift is to divide the value in the register by two (dropping the fractional part). If k = -2 then the LOAD m – load the value at memory location m into the register effect of left shift is to divide the value in the register by four (dropping the fractional part) and so on. The algebraic expression that m Pr Mr implements left shift is

STORE m – store the value in the register at memory location m q q (ln ar) Ur ªNr!

r q Pm Mm q! ªVr!

SHIFT k – shift the value in the register by k bits. If k is positive where the bit shift is to the right and if k is negative the bit shift is to the left. k Ur = Nr - [2 Nr ] The bits are numbered from the leftmost bit which is bit 0 and k Vr = [ 2 Nr ] 61 W = Nr - [ Nr / Nm ] with [ z ] being the value of z truncated to an integer (fractional part and dropped). X = [ Nr / Nm ] ADD m – add the value at memory location m to the value in the register with [ z ] being the value of z truncated to an integer (fractional part dropped). q @ q (ln ar ) Nm ªNr! AND m – change the value in the register by “and-ing” it with the q value at memory location m q! ª(Nr + Nm)!

q q q q @ SUBTRACT m – subtract the value at memory location m from the ((ln ar ) (W) (W) + (ln ar) (-W) (-W) ) ªNr! value in the register (assumes the value in the register is greater than or q equal to the value at location m) q! ªX!

q q where (ln ar) Nm ªNr!

q W = Nr & Nm - Nr q! ª(Nr - Nm)! and where

MULTIPLY m – multiply the value in the register by the value at X = Nr & Nm memory location m with (z) = 1 if z > 0 and 0 if z < 0. The & operator (adopted from the q q @ q C programming language) performs bitwise AND. Corresponding bits (ln ar ) (Nr ) (Nm - 1) ªNr! in each operand are “multiplied” together using the multiplication q rules: q! ª(Nr Nm)!

1 & 1 = 1 DIVIDE m – divide the value in the register by the value at memory 1 & 0 = 0 & 1 = 0 & 0 = 0 location m

q For example the binary numbers 1010 & 1100 = 1000 or in base 10 5 @ q (ln ar ) W ªNr! & 3 = 1. q q! ªX! OR m – change the value in the register by “or-ing” it with the value at memory location m where

62 q q @ replaced by a 0 bit and each 0 bit is replaced by a 1 bit. Since we have (ln ar ) (W) ªNr! infinite words in our computer we supplement this rule with the q q! ªX! restriction that the exchange of 1’s and 0’s only is made up to and including the rightmost 1 bit in the operand. The 0 bits beyond that where remain 0 bits. For example the binary number ~101 = 010 or in base 10, ~3 = 2. W = Nr | Nm - Nr and where INPUT m – input a value storing it at memory location m. The input device is usually associated with a memory location from which the X = Nr | Nm input symbolically takes place. We will designate the memory location The | operator (adopted from the C programming language) performs of the input device as in.

bitwise OR. Corresponding bits in each operand are “multiplied” in together using the multiplication rules: Pm Mm

1 | 1 = 1 | 0 = 0 | 1 = 1 OUTPUT m – output the value at memory location m. The output device is usually associated with a memory location to which output 0 | 0 = 0 symbolically takes place. We will designate the memory location of the output device as out.

For example the binary numbers 1010 | 1100 = 1110 or in base 10 5 | 3 m = 7. Pout Mout

NOT – change the value in the register by “noting” it TRA m – set the value of the program counter to the value at memory location m. If we designate the program counter memory q q q q @ location as pc then this instruction is mapped to ((ln ar ) (W) (W) + (ln ar) (-W) (-W)) ªNr! m q Ppc Mpc q! ªX!

TZR m – set the value of the program counter to the value at where memory location m if the value in the register is zero. W = ~N - N r r (P m M ) (Nr) (-Nr) and where pc pc using (0) = 1.

X = ~Nr

with (z) = 1 if z > 0 and 0 if z < 0. The ~ operator (adopted from the C programming language) performs bitwise NOT. Each 1 bit is 63 HALT – stop execution of the program. The halt in a program is where N is a normalization constant. mapped to a “bra” state vector. This simple program does not produce a superposition of states. As a < … | result programs of this type are analogous to ordinary programs for normal, non-Quantum computers. The numbers in memory after the A Simple Assembly Language Program program concludes are the “output” of the program. We will see Assembly language instructions can be combined to form an assembly programs in succeeding sections that take a computer of fixed state N @ n @ m @ p language program. Perhaps the best way to see how the algebraic (r ) (a0 ) (a1 ) … | 0 > and produce a superposition of states that must representation of assembly language works is to translate a simple be interpreted quantum mechanically. These programs are quantum in assembly language program into its algebraic equivalent. nature and the computer that runs them must be a quantum computer.

The program that we will consider is: Programs and Program Logic The simple program of the last section corresponded to a sequential 1 INPUT x program that executed step by step. We now turn to more complex 2 INPUT y programs with program logic that supports non-sequential execution of 3 LOAD x programs. When this type of program executes the execution of the 4 ADD y instructions can lead to jumps from one instruction to another 5STORE z instruction in another part of the program. 6 OUTPUT z 7 HALT Programs are linear – one instruction executes after another. But they are not sequential – the instructions do not always execute step by step This program translates to the algebraic equivalent: sequentially. A program can specify jumps (“goto” instructions) in the code from the current instruction to an instruction several steps after 7 6 5 4 3 2 1 Steps the current instruction or several steps back to a previous instruction. z r @ Ny x in in < … | Pout Mout Pz Mz (ar ) ªNr! Pr Mr Py My Px Mx | … > The code then executes sequentially until the next jump is

ª(Nr + Ny)! encountered. @ where the power of ar is represented by a power series expansion as These jumps in the code at the level of assembly language implement seen earlier. the control constructs such as goto statements, if expressions, for loops, and switch expressions seen in higher level languages such as C The algebraic expression in the brackets produces one output state and C++. from a given initial state. The values in memory after the last step correspond to one and only one output state of the form: Jumps in code can be implemented in the algebraic representation of programs by having a program counter memory value that increments as the algebraic factor corresponding to each step executes. Steps in @ n @ m @ p @ < n, m, p, … | = (N (r ) (a0 ) (a1 ) … | 0 >) the program can execute or not execute depending on the current value of the program counter.

64 Changes in the program counter value are made using the TRA and 1 INPUT x TZR instructions. In the algebraic representation the program counter 2 INPUT y variable can be used to manage the execution of the program steps. 3 LOAD x 4 TZR y The key algebraic constructs supporting non-sequential program 5 ADD y execution are: 6STORE z 7 OUTPUT z Execute instruction only if PC [ n 8 HALT

( … ) ( n - Npc) This program has the new feature that if the first input – to memory location x – is zero, then instruction 4 will cause a jump to the Execute instruction only if PC m n instruction specified by the value stored at memory location y.

( … ) ( Npc - n) For example if the inputs are 0 placed at memory location x and 2 placed at memory location y, then the TZR instruction will cause the Execute instruction only if PC = n program to jump to instruction 2 from instruction 4. Then the program will proceed to execute from instruction 2. ( … ) ( Npc – n) Another example of a case with a jump is if the input to memory location x is zero and the input to memory location y is 6 then the Execute instruction only if PC not equal to n program jumps from instruction 4 to instruction 6 and the program completes execution from there. If the input to memory location x is ( Npc – n) + ( n - Npc) - 2( Npc – n) ( … ) non-zero no jump takes place. where the parentheses contain one or more instructions and where To establish the algebraic equivalent of the preceding example we (x) = 1 if x = 0 and zero otherwise. The function (x) can be have to use the non-sequential constructs provided earlier in this represented by step functions as section. In addition we must define the equivalent recursively because of the possibility that the program may jump backwards to an earlier (x) = (x) (-x) instruction in the program. If only “forward” jumps were allowed then recursion would not be needed. Using these constructs we can construct non-sequential programs that supprt “goto’s”, if’s and other control constructs seen in higher level An algebraic representation of the program that supports only forward languages. leaps is:

To illustrate this feature of the algebraic representation we will 8 7 @ z (Npc- 7) consider an enhancement of the assembly language program seen < … | ( apc Pout Mout) earlier: 65 5 @ @ Ny (Npc- 5) 6 apc (ar ) ªNr! @ r (Npc- 6) ( ) (apc Pz Mz) ª(Nr + Ny)!

5 @ @ Ny (Npc- 5) 4 apc (ar ) ªNr! @ 1-(Nr) y (Nr) (Npc- 4) ( ) (apc ) (f()Ppc Mpc) ª(Nr + Ny)!

4 3 1- (N ) y (N ) (N - 4) @ r r pc @ x (Npc- 3) (apc ) (Ppc Mpc) ( apc Pr Mr )

3 2 @ x - apc Pr Mr @ in (Npc 2) ( apc Py My)

2 @ in 1 apc Py My @ in (Npc- 1) ( apc Px Mx) 1 @ in The program is apc Px Mx @ @ f()apc Mpc | … > apc Mpc | … > This program is well behaved except if the input value placed at the y The program steps are numbered above each corresponding memory location is 4. In this case the program recursively executes expression. The step function expressions enable the jump to take forever. This defect can be removed by using another memory location place successfully. for a counter variable.

A program with forward and backward jumps supported requires a We can modify the program so that the program only recursively calls recursive definition. We will define the recursive function f() with: itself a finite number of times by having each recursive call decrease the counter variable by one. When the value reaches zero the recursion 7 terminates. An example of such a program (set to allow at most 10 @ z (Npc- 7) f() = (apc Pout Mout) iterations of the recursion) is:

6 7 @ - @ r (Npc- 6) (a z (Npc 7) (apc Pz Mz) g() = pc Pout Mout)

66 6 represent assembly language programs. On the other hand they are @ r (Npc- 6) (apc Pz Mz) quantum in the sense that they use states and harmonic oscillator-like raising and lowering operators. The types of programs we are creating 5 in this approach are “sharp” on the space of states. One input state @ @ Ny (Npc- 5) evolves through the program’s execution to one and only one output apc (ar ) ªNr! ( ) state with probability one. ª(Nr + Ny)! These types of programs are analogous to free field theory in which

4 incoming particles evolve without interaction to an output state @ 1-(Nr) @ 1-(Nw) (Nw) y (Nr) (Npc- 4) (apc ) ((apc ) (g()) awPpc Mpc) containing the same particles.

3 In the next section we extend the ideas in this section to quantum @ x (Npc- 3) ( apc Pr Mr ) programming where a variety of output states are possible – each with a certain probability of being produced. 2 @ in (Npc- 2) Quantum Assembly Language™ Programs ( apc P M ) y y In this section we will first look at a simplified quantum program that illustrates quantum effects but in actuality is a sum of deterministic 1 - assembly language programs mapped to algebraic equivalents. ( a @ in (Npc 1) pc Px Mx) Consider a “quantum” program that is the sum of three ordinary programs g1(), g2() and g3() of the type seen in the last section. Further let us assume the set of orthonormal states The program is | n, m, p, … > @ @ 10 g()apc Mpc (aw ) Mw | … > that we saw in the previous sections with where w is some memory location. We conjecture that any assembly language program using the previously specified instructions can be < X | Y > = XY mapped to an algebraic representation – possibly with the use of additional memory for variables such as the counter variable seen where XY represents a product of Kronecker functions in the above. individual values in memory of the | X > and | Y > states. Further let us assume Using the algebraic constructs supporting non-sequential program execution we can create algebraic representations of assembly | n1, m1, p1, … > = g1()| … > language programs. These programs have a definite input state and through the execution of the program they evolve into a definite output | n2, m2, p2, … > = g2()| … > state -–not a superposition of output states. Therefore they faithfully 67 | n3, m3, p3, … > = g3()| … > | (t) >

for some initial state of the quantum computer. Then to denote the state of a Quantum Computer at time t. The general state of the computer at time t can be written as a superposition of the g1() + g2() + g3()| … > number representation states:

is a “quantum” program where , , and are constants such that | (t) > = fn(t)| n1, n2, n3, … > n ||2 + ||2 + ||2 = 1 where n represents a set of values n1, n2, n3, …

The quantum program produces the state | n1, m1, p1, … > with 2 2 The time evolution of the states can be specified using the Hamiltonian probability || , the state | n2, m2, p2, … > with probability || , and the 2 operator H as state | n3, m3, p3, … > with probability || . | (t) > = e-iHt| (0) > We now have a quantum probabilistic computer. The programs g1(), g2() and g3() are being executed in parallel in a quantum probabilistic With this Hamiltonian formulation we can imagine wishing to manner. simulate a physical (or mathematical) process, defining a Hamiltonian that corresponds to the process, and then creating an experimental Currently, the most feasible way of creating a Quantum Computer setup using a set of lattice spins in some material that implements the with current technology or reasonable extrapolations of current simulation. The experimental setup will prepare the initial state of the technology is to create a material which approximates a lattice with spins, create a fine tuned interaction that simulates the physics of the spins at each lattice site that we can orient electromagnetically at the process, and then, after the system has evolved, will measure the state beginning of a program. The execution of a program takes place by of the system at time t. Repeated performance of this procedure will applying electromagnetic fields that have a time dependence specific determine the probability distribution associated with the final state of to the computation. The electromagnetic fields implement a custom- the Quantum Computer. The probability distribution is specified by tailored set of interactions between the spins in the material that |f (t)|2 as a function of the sets of numbers denoted by n. simulates the calculation to be performed. n A simple example of a Hamiltonian that causes a Quantum Computer The interactions are specified with some Hamiltonian or some to evolve in a non-trivial way is: effective Hamiltonian and the initial state of the lattice spins evolves ∞ dynamically to some configuration that is then measured. @ H = am+1 am m = 0 The Hamiltonians are normally specified using the space-time formalism that is a familiar part of Quantum Mechanics. A (This example was chosen partly because it has a form similar to a Hamiltonian specifies the time evolution of a system starting from an Virasoro algebra generator in SuperString Theory.) Let us assume the initial state. We can introduce an explicit time dependence in states by initial state of the Quantum Computer at t = 0 is using the notation:

68 | 1, 0, 0, 0, … > $ (STORE m) (SUBTRACT “1”)(LOAD m)

that is, an initial value of 1 in the first word in memory and zeroes in where a value is loaded into the register from memory location m and all other memory locations. At time t the state of memory is: then 1 is added to the value in the register. The “1” expression represents a literal value one not a memory location. The parentheses nth memory location around m+1 indicates it is the (m+1)th memory location – not the th addition of one to the value at the m location. ∞ | (t) > = fn(t)| 0, 0, … , 1, 0, … > n = 0 The preceding assembly language expression for H can be replaced with the algebraic representation expression: with n fn(t) = (-it) /n! ∞ r @ m+1 r m H = Pm+1 Mm+1 ar 1 Pr MrPm MmarªNrPr Mr using the power series expansion of the exponentiated Hamiltonian m = 0 ª(Nr + 1) expression. The probability of finding the state This complex expression is not an improvement in one sense. The original Hamiltonian expression was much simpler. Its importance is the mapping that it embodies from a quantum mechanical Hamiltonian nth memory location to an assembly language expression to an algebraic representation of the assembly language. | 0, 0, … , 1, 0, … > is If we regard the value in the register as a “scratchpad” value as @ programmers often do, then we can establish a representation of am n 2 (t /n!) and am in terms of the algebraic representation of assembly language instructions. At first glance the Hamiltonian approach is very different from the @ r @ m Quantum Assembly Language™ approach discussed above. However am h Pm Mm ar 1 Pr Mr these approaches can be interrelated in special cases and (we ª(Nr + 1) conjecture) in the general case through sufficiently clever transformations. For example, the preceding Hamiltonian can be re- and expressed as assembly language instructions r m am h Pm Mm ar ªNr Pr Mr ∞ H = (STORE (m+1))(ADD “1”)(LOAD (m+1)) $ m = 0 The power series expansion of the exponentiated Hamiltonian in the previous example is an example of the use of Perturbation Theory. The direct solution of a problem is often not feasible because of the 69 complexity of the dynamics. Physicists have a very well developed @ theory for the approximate solution of these difficult problems called { ri, rj } = ij Perturbation Theory. Perturbation Theory takes an exact solution of a simplified version of the problem and then calculates corrections to { ri, rj} = 0 that solution that approximate the exact solution of the problem. In the @ @ preceding example the initial state of the Quantum Computer { ri , rj } = 0 represents a time-independent description of the Quantum Computer. The time-dependent description of the Quantum Computer which is The ground state of the computer is the state with the values at all bit the sought-for solution requires the evaluation of the result of the memory locations set to zero. It is represented by the vector application of the exponentiated Hamiltonian to the initial state. For a small elapsed time, the exponential can be expanded in a power series | 0, 0, 0, … > h | 0 > h V and the application of the first few terms of the power series to the initial state approximates the actual state of the Quantum Computer. A typical state of the computer will be represented with a vector such Thus we have a Perturbation Theory for the time evolution of the as Quantum Computer expressed as an expansion in powers of the @ @ @ elapsed time. | 1, 1, 1, … > = r b0 b1 … | 0 >

Bit-Level Quantum Computer Language with the first number being the value in the register, the second In the previous section we examined a Quantum Assembly number the value at memory location 0, the third number the value at Language™ with words consisting of an infinite sets of bits. In this memory location 1, and so on. section we will examine the opposite extreme – a Quantum Computer Language with one-bit words. One can also create Quantum Computer A specified Quantum Computer state evolves as a Quantum Computer Languages for intermediate cases such as 32-bit words. Program executes to a final computer state. A Bit-Level Quantum Computer Program can be represented as an algebraic expression in A Bit-Level Quantum Computer Language can be represented with anti-commuting raising and lowering operators. The approach is @ anti-commuting Fermi operators bi and bi for i = 0, 1, 2, … similar to the approach seen earlier in this chapter for infinite-bit representing each bit location in the Quantum Computer’s memory words using commuting operators. with the anti-commutation rules: Basic Operators of the Bit-Level Quantum Language @ { bi, bj } = ij The key operators that are required for the algebraic representation of a Bit-Level Quantum Computer Language™ are: { bi, bj} = 0 Fetch the Value at a Memory Location (Number Operator) @ @ { bi , bj } = 0 @ Nm = bm bm where ij is 1 if i = j and zero otherwise. We will assume an (unrealistic) one-bit register with a pair of raising and lowering For example, operators r and r@ for the register with the anti-commutation relations: 70 n Nm | … , 1, … > = | … , 1, … > The above expression for Pm is also symbolic. The expression represents the following expression in which the operators are carefully ordered to avoid complications (c-numbers etc.) resulting mth memory location value from reordering.

q Set the Value at Memory Location m to Zero n @ q (ln bm ) Nn Pm h Nm q Mm = (bm) q!

2 The above expression for Mm is symbolic. The expression represents where the sum over q ranges from 0 to ∞. Using the identity Nm = Nm n the following expression in which the operators are carefully ordered the expression for Pm simplifies to: to avoid complications (c-numbers etc.) resulting from reordering. n Pm = 1 + (bm –1)Nm q q Nm ln bm (ln bm) Nm Mm h e = n th When Pm is applied to a state it changes the value of the m memory q q! location from zero to the value at the nth memory location.

th th where the sum ranges from 0 to ∞. Mm becomes m n

Mm = 1 + (bm – 1)Nm

2 n @ x using the identity Nm = Nm . When Mm is applied to a state it sets the Pm | … , 0, … , x, … > = (bm ) | … , 0, … , x, …> value of the mth memory location to zero. = | … , x, … , x, … >

n We can use the operators Mm and Pm to express bit-wise assembly Mm | … , x, … > = | … , 0, … > language instructions:

mth memory location value Change the Value at Memory Location m from 0 to the Value at LOAD m – load the value at memory location m into the register Location n m @ Pr Mr = (1 - Nr + br)(1 - Nm) + (Nr + br )Nm n @ Nn Pm = (bm ) The first term on the right handles the case Nm = 0 and the second term on the right handles the case Nm = 1.

q q @ q @ (Nm -1)Nr (ln br ) (Nr ) (Nm - 1) STORE m – store the value in the register at memory (br ) = location m q q!

r @ Pm Mm = (1 - Nm + bm)(1 - Nr) + (Nm + bm )Nr

= 1 + (br – Nr)(1 - Nm) The first term on the right handles the case Nr = 0 and the second term on the right handles the case Nr = 1. Other assembly language instructions can be expressed in algebraic form as well. ADD m – add the value at memory location m to the value in the register The operator algebra that we have developed for a bit-wise Quantum q (ln b @)q N Assembly Language™ or a Quantum Machine Language™ provides a (b @)Nm = r m r framework for the investigation of the properties of Quantum q q! Languages within an algebraic framework – a far simpler task than the standard quantum linguistic approaches. @ = 1 + (br – 1)Nm Quantum High Level Computer Language Programs If both the register and memory bit m have values of one then the The Quantum Assembly Language™ representation that we have application of this operator expression to the quantum state produces developed earlier in this chapter forms a basis for high level Quantum zero. Programming Languages. These languages are analogous to high level computer languages such as C or C++ or FORTRAN. SUBTRACT m – subtract the value at memory location m from the value in the register In ordinary computation a statement in a high level language such as

q q a = b + c; Nm (ln br) Nm (br) = q q! in C programming is mapped to a set of assembly language by a C compiler. A simple mapping of the above C statement to assembly language would be = 1 + (br – 1)Nm

If the value in the register is zero and the value at location m is one the LOAD ab application of this operator produces zero. ADD ac STORE aa MULTIPLY m – multiply the value in the register by the value at memory location m where aa is the memory address of a, ab is the memory address of b and ac is the memory address of c.

72 If we decide to define a High Level Quantum Computer Language™ then it would be natural to define it analogously in terms of a Quantum h1() + h2() + h3()| … > Assembly Language™. A statement in the High Level Quantum Computer Language™ would map to a set of Quantum Assembly where , , and are constants such that Language™ instructions. ||2 + ||2 + ||2 = 1 For example, a = b + c would map to the algebraic expression The sum of simple programs h1() + h2() + h3() produces the state | r @ Nac ab 2 Paa Maa (ar ) ªNr! Pr Mr n1, m1, p1, … > with probability || , the state | n2, m2, p2, … > with 2 2 ª(Nr + Nac)! probability || , and the state | n3, m3, p3, … > with probability || . using the formalism developed earlier in this chapter to LOAD, ADD An initial eigenstate of the number operators is tranformed into an and STORE. output state that is a superposition of number operator eigenstates. In this case we use probabilities to specify the likelihood that a given The definition of high level Quantum Computer Languages™ in this output eigenstate will be found when the output state is measured. approach is straightforward. One can then imagine creating programs in these languages for execution on Quantum Computers just as A Hamiltonian can also be used to specify the time evolution of a ordinary programs are created for ordinary computers. system starting from an initial state. Using the notation:

Another approach to higher level Quantum Computer Languages™ is | (t) > to simply express them directly using raising and lowering operators – not in terms of Quantum Assembly Language™ instructions. For to denote the state of a Quantum Computer at time t the general state example the preceding a = b + c; statement can be directly expressed of a computer at time t can be written as a superposition of number as representation states: | (t) > = fn(t)| n1, n2, n3, … > @ Nac+Nab Naa n (aaa ) (aaa ) 1

ª(Nac + Nab)! ªNaa! where n represents a set of values n1, n2, n3, …

Simple High Level Quantum Computer programs can be expressed as The time evolution of the states can be specified using the Hamiltonian products of algebraic expressions embodying the statements of the operator H as program. These programs are sharp on the set of memory states taking an initial memory state that is an eigenstate of the set of number | (t) > = e-iHt| (0) > operators Nm into an output eigenstate of the number operators. A simple example of a Hamiltonian that causes a Quantum Computer A general High Level Quantum Computer Program is a sum of simple to evolve in a non-trivial way is: High Level Programs. For example,

∞ the pointer variable ptr. The * operator can also be used to set the @ Nm+1+Nm Nm+2 H = (am+2 ) (am+2) 1 value at a memory location as illustrated by the second * example. In m = 0 this example the value 99 is placed at the memory location (address) ª(Nm+1+Nm)! ªNm+2! specified by the ptr pointer variable.

This Hamiltonian is based on the a = b + c statement above. This These operators can be implemented in the algebraic representation of Hamiltonian generates a complex superposition of states as time the Quantum C Language™ in the following way: evolves. More complex Hamiltonians equivalent to programs with several statements can be easily constructed. & w [A, ]

@ Quantum C Language™ where A = m(am - am ) with the sum from 0 to ∞. If we apply the One of the most important computer languages is the C programming operator to a raising or lowering operator we obtain its address language developed at Bell Laboratories in the 1970’s. The original version of version of the C language was a remarkable combination of @ @ & am = [A, am ] = m = & am = [A, am] low level (assembly language-like) features and high level features like the mathematical parts of FORTRAN. The variables in the The equivalent of the * operator is actually a pair of operator language were integers stored in words just as we saw in the earlier expressions. To fetch the value at a memory location we use examples in this chapter. (There were several other types of integers as well – a complication that we will ignore.) *m h Nm

Using the ideas seen in the earlier sections of this chapter it is easy to To set the value to X at a memory location m we use a more complex develop algebraic equivalents for most of the constructs of the C C language representation: language and thus create a Quantum C Language™. An important element that must be added to the previous development is to *m = X; introduce the equivalent of pointers. Simply put pointers are variables that have the addresses of memory locations as their values. The C An equivalent algebraic expression is: language has two important operators for pointer manipulations: @ X Nm Operator Role Example *m(X) h (am ) (am) 1 & Fetch an address ptr = &x; ªX! ªNm!

* Fetch/set the value at an address z = *ptr; *m(X) is a functional notation. So a = b + c can be rewritten as a *ptr = 99; “pointer” algebraic expression as: The & operator of C fetches the address of a variable in memory. The @ *ac + *ab *aa example shows a pointer variable ptr being set equal to the address of (aaa ) ( a aa ) 1 the x variable. The * (dereferencing) operator can fetch the value at a ª(*ac + *ab)! ª*aa! memory location. The first * example illustrates this aspect: the variable z is set equal to the value at the memory location specified by 74 Or more compactly using the functional notation as

*aa(*ab + *ac)

The Quantum C Language™ could be used to define Hamiltonians for a Quantum Computer. Other languages such as Java™, C++, lisp and so on also have Quantum analogues which may be defined in a similar way.

SuperString Quantum Computer Revisited The concepts we have been exploring for the algebraic representation of Quantum Programs raise the question - Can the SuperString Quantum Computer and Superstring interactions in particular be cast into the form of algebraic quantum computations.

It appears that it may be possible to accomplish this purpose. The basis for this formulation is first the use of the raising and lowering operator formalism as the primary formalism and the relegation of the space- time formalism of SuperStrings to a secondary status. Secondly, the state-operator isomorphism of SuperString theory can be used to develop a representation of vertex operators for interactions as as states within the processor part of the SuperString Quantum Computer. This surprising feature of SuperString theory answers a very basic question confronting the SuperString Quantum Computer: How do we represent SuperString interactions in the Quantum Computer framework. The answer appears to be that we treat interactions (vertex operators) as states within the Quantum Computer. Without this feature of SuperString theory the representation of interactions in the SuperString Quantum Computer would be an open question.